## POLYNOMFAKTORISIERUNG UEBER GLOBALEN FUNKTIONENKOERPERN ##KAPITEL 5 - Anwendungen- ## LAUFZEIT-TABELLE 5.1 FUR DEN ZAHM VERZWEIGTEN FALL ## &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& ## Im Folgenden findet man die Beispiele, die in der Diplomarbeit ## im letzten Kapitel vorkommen. Die folgende Polynome wurden in Kant/Kash3 ## faktorisiert. Bevor man folgende Beispiele berechnet, muss man ## zuerst mit dem "Read" Kommand in Kash3 das fu"r die Diplomarbeit implementierte ## Programm aufrufen. ## 09.06.2011, Berlin, Muenevver Akay ## Technischen Universitaet von Berlin ########################################################## ########################################################## #### #### 1. BEISPIEL ########################################################## ########################################################## k := GF(7,2); ## Endlicher Koerper kx := PolynomialAlgebra(k); ## Polynomalgebra von k kT := RationalFunctionField(k); ## Rationaler Funktionen Koerper kTy := PolynomialAlgebra(kT); ## Polynomalgebra kT AssignNames_ (kTy, ["y"]); ## Damit auf dem Bildschirm bei der Ausgabe nur 'y' erscheint y := kTy.1; ## Damit man bei der Eingabe statt 'kTy.1' nur 'y' schreiben kann AssignNames_ (kT, ["T"]); ## Damit auf dem Bildschirm nur 'T' erscheint T := kT.1; ## Damit man ibei der Eingabe statt 'kT.1' nur 'T' schreiben kann f := y^5+2*y^2*T+2*y*T+T+1;; ## Funktionenkoerper erzeugendes Polynom F := FunctionField(f); ## Funktionenkoerper o := MaximalOrderFinite(F); ## Maximalordnung ot := PolynomialAlgebra(o); ## Polynomalgebra von o t := ot.1; ## Damit man bei der Eingabe statt 'ot.1' nur 't' schreiben kann AssignNames_ (ot, ["t"]); ## Damit auf dem Bildschirm nur 't' erscheint ## Das Polynom g aus o_{K}[t] hat folgende Faktoren: ## ************************************************* h1 :=t+Coerce(o,[T+1,2*T,T,0,0]); h2 :=t-Coerce(o,[1,T,T,0,T]); h3 :=t+Coerce(o,[T^2+1,1,T^3,1,0]); h4 := t^3+Coerce(o,[3,6,T+1,T+1,1]); h5 := t^2+Coerce(o,[T^2+1,2*T+1,4*T,1,6+T]); h6 := t^3+Coerce(o,[T^3,T^7+2,2*T^2,0,0]); ## Das Polynom g erhaelt man durch: ##******************************** g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F); ## Damit erhaelt man den Grad von g und die hoechste Potenz aller Koeffizienten von g ##[ 11, 18 ] Das Ergebnis von Coefgr(g,f,F); Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ## Faktorisierungsalgorithmus ## [ [ t + [ T + 1, 2*T, T, 0, 0 ] ], 3, 184, 1 ] Mit der Reinfolge: Der gefundene Faktor, Durchgefuehrte Hensel Schritte, Abschaetzung fuer die Koeffizienten der potenziellen Faktoren von g, Die Reihe des Listenelements bei der Faktorisierung ueber Restklassenkoerper. ## Time: 8604.68449 s Dauer des Algorithmus insgesamt mit einige Kommentare. Das Polynom g hat folgende Gestalt: *********************************** t^11 + [ T^2 + T + 1, T + 1, T^3, 1, 6*T ]*t^10 + [ 3*T^3 + 4*T^2 + T, T^5 + T^4 + 6*T^3 + 2*T^2 + T + 1, 2*T^5 + T^4 + 6*T^3 + 4*T, 2*T^5 + T^4 + 2*T^3 + 6*T^2 + T + 1, 6*T^3 + 6 ]*t^9 + [ 5*T^7 + T^6 + 5*T^5 + T^4 + 2*T^3 + 4*T^2 + T + 4, 4*T^7 + T^6 + 2*T^4 + T^3 + 6*T^2 + T + 1, 3*T^7 + 5*T^6 + 2*T^5 + 4*T^4 + 3*T^3 + T^2 + 5*T + 3, 2*T^6 + 2*T^5 + T^4 + 2*T^3 + 4*T^2 + 2*T + 2, 6*T^6 + 4*T^5 + 5*T^3 + 2*T^2 + 2 ]*t^8 + [ T^9 + T^8 + 5*T^7 + 4*T^6 + 6*T^5 + 4*T^3 + 3*T^2 + T, 3*T^9 + T^8 + 3*T^7 + 3*T^6 + 3*T^5 + 6*T^4 + 3*T^3 + 4*T^2 +3*T + 2, 2*T^9 + T^8 + 6*T^6 + 6*T^5 + 6*T^4 + 5*T^3 + 6*T^2 + T + 1, T^10 + T^7 + 2*T^6 + 6*T^4 + T^2 + 2*T + 4, 4*T^7 + 3*T^5 + 4*T^4 + 5*T^3 + 2*T^2 + 2*T + 4 ]*t^7 + [ T^11 + T^10 + 3*T^8 + 2*T^7 + T^6 + 3*T^5 + 6*T^4 + 2*T^3 + 3*T^2 + 3*T + 4, 2*T^11 + 3*T^10 + T^9 + 5*T^8 + T^7 + 2*T^6 + 4*T^5 + 4*T^4 + 4*T^3 + T^2 + 5*T + 2, T^12 + 3*T^11 + 6*T^10 + 3*T^9 + T^8 + 4*T^7 + 5*T^6 + 5*T^5 + 2*T^3 + 5*T^2 + 6*T + 6, 2*T^12 + T^11 + 6*T^10 + T^9 + 6*T^8 + T^7 + 5*T^6 + 4*T^5 + 3*T^4 + 4*T^3 + 5*T^2 + 5*T + 3, 2*T^12 + T^11 + 2*T^10 + 6*T^9 + T^8 + 6*T^7 + 2*T^6 + 3*T^5 + T^4 + 4*T^2 + 6*T + 5 ]*t^6 + [ T^14 + 4*T^13 + 3*T^12 + 2*T^11 + 5*T^10 + 6*T^9 + 5*T^8 + 4*T^7 + 3*T^6 + 6*T^5 + T^4 + T^3 + 6*T^2 + 4*T + 4, 5*T^12 + 5*T^11 + 5*T^8 + 3*T^6 + 5*T^5 + 4*T^4 + 5*T^2 + 2*T + 3, 5*T^14 + 6*T^11 + 4*T^9 + T^8 + 5*T^5 + 3*T^4 + T^3 + T^2 + 5*T + 3, 3*T^14 + 5*T^13 + 2*T^12 + 4*T^11 + 3*T^10 + 4*T^9 + 2*T^8 + 5*T^7 + 6*T^6 +2*T^5 + 6*T^4 + T^2 + 6*T, 2*T^13 + 2*T^12 + T^11 + 2*T^10 + 2*T^9 + 3*T^8 + T^7 + T^6 + 2*T^5 + 3*T^4 + 2*T^3 + 5*T^2 + 5*T + 2 ]*t^5 + [ 4*T^15 + 4*T^14 + 6*T^13 + T^12 + 4*T^11 + 3*T^10 + 5*T^8 + T^7 + T^5 + 3*T^4 + T^3 + 4*T^2 +3*T + 3, T^15 + 5*T^14 + 2*T^13 + 2*T^12 + 3*T^11 + 3*T^9 + 5*T^8 + 5*T^7 + 6*T^6 + 5*T^5 + 2*T^4 + 5*T^3 + 6*T^2 + 5*T + 5, T^15 + 2*T^14 + T^13 + 4*T^10 +2*T^9 + 6*T^8 + 5*T^7 + 5*T^6 + 5*T^5 + 4*T^4 + 2*T^3 + 3*T^2 + 2*T, 6*T^14 +5*T^13 + 6*T^12 + 3*T^10 + 6*T^9 + T^8 + 6*T^7 + 2*T^4 + 2*T^3 + 3*T^2 + 2*T + 6, T^14 + 2*T^13 + 2*T^11 + 5*T^10 + 5*T^9 + 4*T^7 + 6*T^6 + 6*T^5 + 6*T^4 +4*T^3 + 4*T^2 + 3 ]*t^4 + [ 6*T^15 + 5*T^14 + 5*T^13 + 5*T^12 + 4*T^10 + 5*T^9 + 2*T^8 + 5*T^6 + 6*T^4 + 4*T^3 + 2*T^2 + T + 6, 4*T^15 + 5*T^14 + 2*T^13 + 3*T^12 + T^11 + 4*T^9 + 3*T^8 + 5*T^7 + 3*T^6 + T^5 + 6*T^3 + T^2 + T + 1, 4*T^16 + 2*T^14 + 2*T^12 + 2*T^11 + 5*T^10 + 6*T^9 + 6*T^7 + T^6 + 3*T^4 + 5*T^3+ T^2, T^16 + 4*T^15 + 6*T^14 + 3*T^13 + T^12 + 6*T^11 + 5*T^10 + 2*T^9 + 5*T^8 + T^7 + 2*T^6 + 5*T^5 + 4*T^4 + T^2 + 3*T + 2, T^16 + 5*T^14 + 4*T^13 + 3*T^12 + 5*T^11 + 4*T^10 + 5*T^9 + T^8 + 6*T^6 + 3*T^5 + 6*T^4 + 3*T^3 + 2*T^2 + 5*T ]*t^3 + [ 6*T^16 + 4*T^15 + 5*T^13 + 4*T^12 + 3*T^11 + 3*T^10 + 3*T^9 + 4*T^8 + 2*T^6 + T^5 + 4*T^3 + 2*T + 4, 2*T^16 + 6*T^15 + 6*T^14 + 6*T^13 + 3*T^12+ 6*T^11 + T^10 + 3*T^9 + 4*T^8 + 5*T^7 + 5*T^6 + 3*T^5 + 4*T^3 + 3*T + 2, 6*T^16 + 4*T^15 + 3*T^14 + 5*T^13 + 4*T^11 + T^10 + 6*T^9 + 4*T^8 + 6*T^7 + 3*T^6 + 5*T^5 + 5*T^4 + T^3 + 3*T + 4, T^16 + 3*T^15 + 2*T^14 + 2*T^13 + 6*T^9 + 6*T^8 + 5*T^7 + 5*T^6 + 2*T^5 + T^4 + 5*T^3 + T^2 + T + 3, 2*T^15 + 2*T^14 + 3*T^13 + T^11 + 6*T^10 + T^8 + T^7 + 6*T^6 + 6*T^5 + T^4 + 2*T^3 + 6*T^2 + 6*T +2 ]*t^2 + [ 6*T^17 + 5*T^16 + 4*T^15 + T^14 + 6*T^13 + 2*T^12 + 4*T^10 + 5*T^8 + 3*T^7 + 5*T^5 + 4*T^4 + 2*T^3 + 4*T^2 + 4*T + 3, 5*T^17 + 4*T^16 + 3*T^15+ 6*T^14 + 4*T^13 + 4*T^12 + T^11 + 6*T^10 + 5*T^8 + 2*T^7 + 5*T^6 + T^5 + T^4+ 6*T^3 + 3*T^2, 5*T^17 + 6*T^16 + 3*T^15 + 4*T^14 + 4*T^13 + 3*T^12 + 2*T^11+ 4*T^10 + 6*T^9 + 5*T^8 + 4*T^7 + 3*T^6 + 2*T^5 + 4*T^3 + 5*T^2 + T + 3, T^16 + 4*T^15 + 5*T^14 + 5*T^13 + 2*T^12 + 6*T^11 + 6*T^9 + 6*T^8 + 6*T^7 + T^6 +2*T^5 + 4*T^4 + 4*T^3 + 5*T^2 + 2*T, 2*T^13 + 4*T^12 + T^11 + 3*T^9 + 4*T^8 +6*T^7 + 6*T^6 + 2*T^5 + T^4 + 5*T^3 + 6*T^2 + T + 2 ]*t + [ 4*T^18 + 3*T^17 +2*T^16 + T^15 + 5*T^14 + 3*T^11 + 2*T^10 + T^9 + 4*T^8 + 2*T^7 + 2*T^6 + 6*T^5 + 6*T^4 + 3*T^3 + 4*T + 5, 6*T^18 + 6*T^15 + T^14 + T^12 + 2*T^11 + 6*T^10 +4*T^8 + 2*T^7 + 5*T^6 + 5*T^5 + 4*T^4 + T^3 + 3*T^2 + 5*T + 3, 3*T^18 + 3*T^17 + 3*T^16 + 2*T^15 + 2*T^12 + 3*T^10 + 3*T^9 + 2*T^8 + T^7 + 6*T^6 + T^4 + 4*T^3 + 6*T^2 + 4*T + 1, T^18 + 3*T^17 + 4*T^16 + T^15 + 4*T^14 + 5*T^13 + 5*T^11 + 3*T^10 + 3*T^9 + 5*T^8 + T^7 + 3*T^6 + T^5 + 5*T^4 + 5*T^3 + 6, 5*T^18 + 5*T^17 + 5*T^16 + T^15 + T^13 + 5*T^9 + T^8 + T^7 + 2*T^5 + T^4 + T^3 + 3*T^2 +6*T+2 ] #################################################### #################################################### #### #### 2. BEISPIEL #################################################### #################################################### k := GF(7,4); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^3+2*y^2*T^2+y*T^11+T^2+T+21;; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T^2,T+1,2*T+2]); h2 :=t-Coerce(o,[T+1,T^3+1,T+2]); h3 :=t^2+Coerce(o,[1,T,2+T^2]); h4 := t+Coerce(o,[T+1,2,T^6]); h5 := t-Coerce(o,[T^2+T,T^3+1,T^2+6]); h6 := t^3+Coerce(o,[1+T,T^2,2+T]); h7 := t+Coerce(o,[1,2,T]); g := h1*h2*h3*h4*h5*h6*h7;; Coefgr(g,f,F);[10,82] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ********************************** [ [ t + [ 6*T^2 + 6*T, 6*T^3 + 6, 6*T^2 + 1 ] ], 5, 459, 1 ] Time: 9988.644227 s Das Polynom g hat folgende Gestalt: *********************************** t^10 + [ 6*T + 1, 5*T^3 + T + 3, T^6 + 6*T^2 + 2*T + 1 ]*t^9 + [ 5*T^12 + 4*T^11 + T^10 + T^9 + 5*T^8 + T^7 + 5*T^6 + 6*T^5 + 5*T^4 + 2*T^3 + 6*T^2 + 4*T, 5*T^21 + 6*T^20 + 2*T^19 + 6*T^18 + 6*T^17 + 2*T^16 + 3*T^15 + 3*T^14 + 3*T^13 + T^10 + 6*T^9 + 4*T^8 + 6*T^7 + T^5 + 3*T^4 + T^3 + 5*T^2 + 6*T + 5, T^19 + 5*T^18 + 6*T^17 + 2*T^14 + T^13 + 2*T^11 + 4*T^10 + 5*T^9 + 5*T^8 + 2*T^7 + 6*T^5 + 3*T^4 + 4*T^3 + 3*T^2 ]*t^8 + [ 2*T^24 + T^23 + 4*T^22 + 5*T^21 + 3*T^20 + 3*T^18 + 6*T^17 + 5*T^16 + 6*T^15 + 6*T^14 + T^13 + 6*T^12 + 3*T^11 + 3*T^10 + 5*T^9 + 2*T^8 + 6*T^5 + T^4 + 3*T^3 + 3*T^2 + T + 1, 2*T^33 + 6*T^32 + 5*T^31 + 3*T^29 + 4*T^28 + 6*T^27 + 5*T^25 + T^24 + 5*T^23 + T^22 + 2*T^20 + 2*T^19 + 5*T^18 + T^17 + 5*T^16 + 4*T^15 + 3*T^14 + 4*T^13 + 5*T^12 + 5*T^9 + 2*T^8+ 2*T^7 + 3*T^5 + 4*T^4 + 6*T^3 + 5*T^2 + 5*T + 5, 5*T^31 + 6*T^30 + 3*T^29 + 3*T^28 + T^26 + 4*T^25 + 4*T^24 + 2*T^23 + T^21 + 3*T^20 + 4*T^19 + 4*T^16 + T^13 + 3*T^12 + 2*T^11 + 2*T^10 + T^9 + T^7 + 3*T^6 + 5*T^5 + T^4 + 5*T^3 + T^2 + 5*T + 5 ]*t^7 + [ 3*T^36 + 5*T^35 + 4*T^34 + T^33 + 3*T^31 + 2*T^29 + 2*T^28 + 3*T^27 + 4*T^26 + T^24 + 5*T^23 + 3*T^21 + 4*T^20 + 3*T^19 + 4*T^18 + 6*T^17 + 3*T^16 + 3*T^15 + 6*T^14 + 5*T^12 + T^11 + 4*T^9 + 6*T^8 + 6*T^7 + 6*T^6 + 3*T^5 + 3*T^4 + T^3 + T, 3*T^45 + 2*T^44 + 2*T^43 + 6*T^42 + T^41 + 2*T^40 + 5*T^39 + 4*T^38 + 5*T^37 + 5*T^36 + 6*T^35 + 2*T^33 + 4*T^32 + T^31 + T^30 + 5*T^29 + 6*T^28 + 3*T^27 + 4*T^26 + 3*T^24 + 3*T^23 + T^22 + 4*T^21 + 6*T^19 + 5*T^18 + 6*T^15 + 4*T^14 + 2*T^13 + 6*T^12 + 3*T^11 + 2*T^10 + 5*T^9 + 4*T^8 + 5*T^7 + T^6 + 5*T^5 + 5*T^4 + T^3 + 3*T^2 + 6*T + 1, 6*T^43 + 3*T^42 + 3*T^41 + 3*T^40 + 4*T^39 + 5*T^38 + T^37 + 4*T^36 + T^35 + 6*T^34 + 4*T^33 + 2*T^32 + 2*T^31 + 5*T^29 + 4*T^28 + 4*T^27 + T^26 + 2*T^25 + T^24 + 4*T^23 + T^22 + T^21 + 3*T^19 + 5*T^18 + 4*T^17 + 3*T^16 + 3*T^12 + 6*T^11 + 3*T^10 + T^9 + 6*T^8 + 3*T^7 + T^6 + 2*T^5 + 5*T^3 + 6*T^2 + 5*T + 4 ]*t^6 + [ 5*T^47 + T^46 + 6*T^45 + 3*T^44 + 5*T^43 + T^41 + T^39 + 2*T^37 + 3*T^35 + 4*T^33 + 5*T^32 + 4*T^31 + 3*T^30 + 6*T^29 + 5*T^28 + 4*T^26 + 2*T^25 + 2*T^24 + 3*T^22 + 2*T^21 + T^20 + T^19 + 6*T^18 + 6*T^17 + 4*T^16 + 4*T^15 + 3*T^13 + 6*T^12 + 4*T^11 + 4*T^9 + 2*T^8 + 5*T^7 + 2*T^6 + 2*T^5 + 3*T^4 + 5*T^3 + 3*T^2 + 5*T + 6, 5*T^56 + 3*T^55 + 3*T^54 + 5*T^52 + 2*T^51 + 6*T^50 + T^49 + 4*T^46 + 6*T^45 + 6*T^44 + 5*T^43 + 6*T^42 + 6*T^41 + T^40 + T^39 + 5*T^37 + 6*T^36 + 3*T^35 + 4*T^34 + 4*T^33 + 2*T^32 + 5*T^31 + 6*T^30 + 6*T^29 + 2*T^28 + 2*T^27 + T^26 + 2*T^25 + 2*T^23 + T^21 + 6*T^20 + 4*T^19 + 4*T^18 + 3*T^17 + 3*T^16 + T^15 + 3*T^14 + 6*T^13 + 5*T^12 + T^11 + 6*T^10 + 6*T^9 + 2*T^8 + 6*T^7 + 4*T^6 + T^5 + 5*T^3 + T^2 + 3*T, 2*T^55 + 6*T^54 + 2*T^53 + T^52 + 3*T^51 + 6*T^48 + 6*T^46 + 6*T^45 + 5*T^44 + 5*T^43 + T^42 + T^41 + 3*T^40 + 4*T^39 + 2*T^38 + 6*T^37 + T^36 + 2*T^35 + 3*T^34 + 2*T^33 + 3*T^32 + 4*T^31 + 2*T^30 + 2*T^29 + 6*T^28 + 4*T^27 + T^26 + 4*T^25 + 6*T^24 + T^23 + T^22 + 5*T^21 + 3*T^20 + 6*T^19 + 6*T^18 + 4*T^17 + 6*T^16 + 5*T^15 + 2*T^14 + 5*T^13 + 6*T^12 + 5*T^11 + 4*T^10 + 2*T^9 + 4*T^8 + 6*T^6 + 6*T^5 + 2*T^4 + 6*T^3 + 5*T^2 + 2*T + 6 ]*t^5 + [ 2*T^49 + 6*T^48 + 5*T^47 + 3*T^46 + 3*T^44 + 6*T^43 + 3*T^42 + 2*T^41 + 6*T^39 + T^38 + 2*T^37 + T^36 + 2*T^35 + 5*T^34 + 2*T^33 + 6*T^32 + 4*T^31 + 5*T^30 + T^29 + 6*T^28 + 4*T^27 + 5*T^24 + 6*T^23 + 2*T^21 + 2*T^20 + 4*T^18 + T^17 + 4*T^16 + 5*T^15 + T^14 + 6*T^13 + 4*T^12 + T^11 + 5*T^9 + 3*T^8 + 3*T^7 + 5*T^6 + T^5 + 4*T^4 + 5*T^2 + 3*T, 2*T^58 + 4*T^57 + T^56 + 2*T^55 + 5*T^54 + 5*T^53 + T^52 + 2*T^51 + 6*T^48 + 3*T^47 + 5*T^46 + 4*T^45 + 5*T^44 + 6*T^43 + 5*T^42 + 2*T^41 + T^40 + 6*T^39 + 3*T^38 + 4*T^37 + 2*T^36 + 5*T^35 + 5*T^34 + 2*T^33 + 6*T^32 + T^30 + 4*T^29 + 3*T^28 + T^27 + T^26 + T^25 + 4*T^24 + T^22 + 6*T^21 + 3*T^20 + 6*T^19 + T^18 + T^17 + T^16 + 6*T^15 + 4*T^14 + 4*T^12 + 2*T^11 + 5*T^10 + 6*T^9 + 2*T^8 + 2*T^7 + 3*T^6 + 6*T^5 + 4*T^4 + 2*T^3 + 3*T^2 + 4*T + 5, T^56 + 4*T^55 + 6*T^54 + 5*T^53 + 4*T^52 + 3*T^51 + 5*T^50 + 4*T^49 + 5*T^48 + 2*T^47 + T^46 + 4*T^45 + 4*T^44 + 2*T^43 + 5*T^42 + 5*T^40 + T^39 +6*T^38 + 4*T^37 + T^36 + T^35 + 5*T^33 + 5*T^32 + 3*T^31 + T^30 + 2*T^29 + 6*T^28 + 4*T^27 + T^26 + 2*T^25 + 6*T^23 + 5*T^21 + T^20 + 4*T^19 + 4*T^18 + 5*T^17 + 5*T^16 + 3*T^15 + 4*T^14 + 5*T^13 + 2*T^12 + 4*T^11 + 4*T^10 + T^9 + 6*T^8 + 4*T^7 + 5*T^6 + T^4 + T^2 + 2*T + 1 ]*t^4 + [ 4*T^61 + 4*T^60 + 2*T^59 + 6*T^58 + 6*T^57 + 2*T^54 + 2*T^53 + 5*T^52 + 3*T^50 + 4*T^49 + 4*T^48 + 3*T^47 + 4*T^46 + 6*T^45 + T^44 + T^43 + 6*T^42 + T^41 + T^40 + 6*T^39 + T^38 + 5*T^35 + T^34 + 6*T^33 +3*T^32 + 2*T^31 + 3*T^30 + 4*T^29 + 6*T^28 + T^27 + 3*T^26 + 2*T^25 + 6*T^24 + 5*T^23 + T^21 + 5*T^20 +3*T^19 + 5*T^18 + 2*T^17 + 4*T^16 + 3*T^15 + 5*T^14 + T^13 + 5*T^12 + 5*T^11 + 6*T^10 + 3*T^9 + 4*T^7 +5*T^6 + 2*T^5 + 2*T^4 + 6*T^3 + 6*T^2 + 4*T + 6, 4*T^70 + 2*T^68 + 4*T^67 + 2*T^66 + 5*T^65 + 2*T^64 + 2*T^62 + 3*T^61 + 4*T^60 + 4*T^59 + 4*T^58 + T^57 + 6*T^56 + T^55 + 4*T^54 + T^53 + T^52 + T^51 + 4*T^50 + 4*T^49 + 4*T^48 + 6*T^46 + 4*T^45 + T^44 + T^43 + 4*T^42 + 3*T^41 + 4*T^40 + 6*T^39 + 5*T^38 + 5*T^37 + 5*T^36 + 6*T^35 + 6*T^34 + 5*T^33 + T^32 + 2*T^31 + T^30 + 4*T^29 + 2*T^28 + 5*T^27 + 3*T^25 + 6*T^23 + T^22 + 3*T^21 + 3*T^20 + 3*T^19 + 3*T^18 + T^17 + T^16 +4*T^15 + T^14 + 6*T^13 + 5*T^12 + 3*T^11 + 4*T^10 + 6*T^9 + T^8 + 4*T^7 + 2*T^4 + 5*T^3 + 5, 5*T^68 + T^67 + T^66 + T^65 + 5*T^63 + T^62 + 2*T^61 + 2*T^58 + T^57 + 4*T^56 + 4*T^55 + 2*T^54 + 6*T^52 + 4*T^51 + 3*T^50 + 3*T^48 + 5*T^47 + 6*T^46 + 4*T^45 + 3*T^44 + 6*T^43 + 4*T^42 + 4*T^41 + 3*T^40 + 6*T^39 + T^37 + 5*T^36 + T^35 + T^34 + 6*T^33 + 6*T^32 + 4*T^31 + T^29 + 3*T^28 + 5*T^27 + T^26 + 5*T^25 + 5*T^24 + 5*T^23 + T^22 + 6*T^21 + T^20 + 5*T^19 + 3*T^17 + 6*T^15 + 2*T^14 + 2*T^10 + T^9 + 4*T^7 + 4*T^6 + 5*T^5 + 3*T^4 + 4*T^3 + 3*T ]*t^3 + [ 4*T^60 + 3*T^59 + T^58 + T^57 + 2*T^56 + 6*T^54 + 4*T^53 + T^51 + 5*T^50 + 5*T^49 + 5*T^48 + 2*T^47 + T^46 + 4*T^45 + 5*T^42 + 6*T^41 + 6*T^39 + 3*T^38 + 2*T^37 + 2*T^36 + 2*T^34 + 3*T^33 + 5*T^32 + 4*T^30 + 3*T^29 + 3*T^28 + 3*T^26 + 3*T^25 + T^24 + 2*T^23 + 2*T^22 + T^21 + 2*T^20 + 6*T^19 + 3*T^18 + 6*T^17 + 4*T^16 + 5*T^15 + 4*T^14 + T^13 + 5*T^12 + 6*T^11 + 6*T^10 + 6*T^9 + 5*T^8 + 4*T^7 + 4*T^6 + 6*T^4 + 5*T^3 + 6*T^2 + 6*T + 6, 4*T^69 + 6*T^68 + 2*T^67 + 6*T^66 + 3*T^65 + 4*T^64 + 2*T^63 + 2*T^62 + 5*T^61 + 3*T^60 + 2*T^59 + T^58 + 4*T^57 + 4*T^56 + 3*T^55 + 4*T^52 + 3*T^51 + 4*T^49 + T^48 + T^47 + 4*T^46 + 2*T^45 + 4*T^44 + 2*T^43 + 2*T^41 + 3*T^40 + 3*T^39 + 6*T^38 + 4*T^37 + 3*T^36 + 5*T^35 + 6*T^34 + 3*T^33 + T^32 + 4*T^31 + 5*T^30 + T^29 + T^27 + 2*T^26 + 6*T^25 + 5*T^23 + 6*T^22 + 2*T^21 + 3*T^20 + 6*T^19 + 5*T^18 + 5*T^14 + 5*T^13 + 3*T^12 + T^11 + 5*T^10 + 5*T^9 + 3*T^8 + 4*T^6 + 2*T^5 + 4*T^4 + 5*T^3 + 3*T^2 + 3*T + 2, 5*T^67 + 4*T^66 + 2*T^64 + 2*T^63 + T^62 + 2*T^60 + 3*T^58 + 4*T^57 + T^56 + T^55 + 6*T^54 + 6*T^53 + T^52 + T^50 + 2*T^49 + 3*T^48 + 3*T^47 + 2*T^46 + 2*T^45 + 4*T^44 + 3*T^43 + T^42 + T^38 + 3*T^37 + 4*T^34 + 6*T^33 + T^32 + 6*T^31 + T^30 + 3*T^29 + T^28 + T^26 + 6*T^25 + 5*T^24 + 3*T^22 + 6*T^21 + 4*T^20 + 4*T^19 + 2*T^18 + T^17 + 5*T^16 + T^15 + 3*T^14 + 6*T^13 + 3*T^12 + 5*T^11 + 6*T^10 + 5*T^9 + 5*T^8 + 6*T^7 + 3*T^6 + 5*T^5 + 2*T^4 + 6*T^3 + 5*T^2 + 4 ]*t^2 + [ 3*T^60 + 4*T^59 + 4*T^58 + 3*T^57 + 3*T^56 + 4*T^55 + 6*T^54 + 2*T^53 + 2*T^52 + 6*T^49 + 5*T^48 + 6*T^47 + T^46 + 4*T^44 + 5*T^43 + 4*T^42 + 6*T^41 + 3*T^40 + 5*T^39 + 2*T^38 + 2*T^37 + T^35 + 6*T^34 + 4*T^33 + 2*T^32 + 2*T^31 + 3*T^30 + T^29 + 3*T^28 + 2*T^27 + 5*T^26 + 4*T^25 + 3*T^24+ 4*T^23 + T^22 + 2*T^20 + T^19 + 2*T^18 + 4*T^17 + 6*T^16 + 6*T^15 + 2*T^13 + 5*T^12 + 6*T^11 + T^10 + T^9 + T^8 + 2*T^7 + 2*T^6 + 6*T^5 + 2*T^4 + 5*T^3 +5*T^2 + 6*T, 3*T^69 + T^68 + 3*T^67 + 3*T^65 + T^64 + 5*T^63 + 4*T^62 + 5*T^61 + 2*T^60 + 4*T^59 + T^58 + 5*T^57 + 6*T^56 + 3*T^55 + 4*T^54 + 2*T^53 + 5*T^52 + 5*T^51 + 6*T^50 + T^49 + 6*T^48 + 5*T^47 + 4*T^46 + 6*T^45 + T^44 + 4*T^42 + 2*T^41 + 5*T^40 + 2*T^39 + 4*T^38 + 6*T^37 + 6*T^36 + 4*T^35 + 6*T^34 + T^33 + 6*T^32 + 3*T^31 + 3*T^30 + 4*T^29 + 3*T^28 + 6*T^27 + 2*T^26 + 3*T^25 + 4*T^24 + 6*T^23 + 5*T^22 + 6*T^21 + 5*T^20 + 4*T^19 + T^18 + 2*T^16 + 2*T^15 + 2*T^14 + 5*T^13 + T^12 + 6*T^11 + 5*T^10 + 3*T^9 + 4*T^8 + 2*T^7 + T^6 + 5*T^5+ 5*T^3 + 6*T^2 + 4*T, 6*T^68 + T^67 + 4*T^65 + 3*T^63 + 3*T^62 + 2*T^60 + T^59 + 2*T^58 + 4*T^57 + 4*T^56 + 2*T^55 + T^54 + 4*T^53 + T^52 + 5*T^51 + 2*T^50 + 2*T^49 + 3*T^48 + 6*T^47 + 5*T^45 + T^43 + 2*T^41 + 5*T^40 + 2*T^39 + 3*T^37 + 4*T^36 + 6*T^35 + T^34 + 3*T^32 + 5*T^30 + 3*T^29 + 4*T^28 + 6*T^27 + 4*T^26 + 3*T^25 + T^22 + 4*T^21 + 2*T^20 + 5*T^19 + 4*T^18 + 3*T^17 + 4*T^16 + 3*T^13 + T^12 + 6*T^11 + 4*T^10 + 3*T^9 + 6*T^8 + T^6 + 3*T^5 + 4*T^4 + 5*T^3 + 4*T^2 + 5*T + 5 ]*t + [ 6*T^73 + T^72 + 5*T^71 + 4*T^69 + T^68 + 6*T^67 + T^66+ 4*T^64 + T^63 + T^62 + 3*T^61 + 2*T^60 + 4*T^59 + 3*T^58 + 2*T^57 + 5*T^56 + 2*T^55 + 2*T^54 + 4*T^52 + 6*T^50 + 3*T^49 + 6*T^48 + T^47 + 2*T^46 + 4*T^45+ 6*T^42 + 4*T^41 + 4*T^40 + T^39 + 3*T^38 + 4*T^37 + 6*T^35 + T^34 + 6*T^31 + 2*T^30 + 3*T^29 + 5*T^28 + T^27 + 5*T^26 + 4*T^24 + 2*T^22 + 4*T^20 + T^19 +3*T^18 + 2*T^17 + 5*T^15 + 3*T^14 + 2*T^13 + 6*T^12 + 5*T^11 + 6*T^10 + 2*T^9+ 4*T^7 + T^5 + 2*T^4 + 2*T^3 + 4*T^2 + T, 6*T^82 + 2*T^81 + 3*T^80 + 4*T^79 + T^77 + 5*T^76 + 3*T^75 + 4*T^74 + T^72 + 2*T^71 + T^70 + 6*T^69 + 6*T^68 + 4*T^66 + 6*T^65 + 4*T^63 + 6*T^62 + 5*T^61 + 6*T^60 + 3*T^59 + 4*T^58 + 5*T^57 + 3*T^56 + 3*T^55 + 5*T^54 + T^53 + 6*T^52 + 4*T^51 + T^50 + 4*T^48 + 4*T^47 +4*T^46 + 3*T^45 + 2*T^43 + T^42 + 6*T^41 + 5*T^40 + 2*T^39 + 5*T^38 + T^37 + 6*T^36 + 6*T^35 + T^34 + 2*T^33 + 2*T^32 + 4*T^31 + 2*T^30 + 5*T^29 + 2*T^28 + 4*T^27 + 2*T^26 + 6*T^25 + 6*T^24 + T^23 + 4*T^22 + 2*T^21 + 6*T^20 + 5*T^18 + 3*T^17 + T^16 + 4*T^15 + 5*T^14 + 4*T^12 + 4*T^11 + T^10 + 5*T^9 + 3*T^7 + 4*T^6 + 5*T^5 + 3*T^4 + 3*T^3 + 2*T^2 + 5*T, 2*T^80 + 3*T^79 + 4*T^78 + 4*T^77 + 5*T^76 + 2*T^75 + 3*T^74 + T^73 + 4*T^72 + 6*T^71 + T^70 + 4*T^69 + 4*T^68 +6*T^67 + T^66 + T^65 + 2*T^64 + 2*T^63 + T^62 + T^61 + 2*T^60 + 6*T^59 + T^58+ 6*T^56 + 2*T^55 + 6*T^54 + 2*T^52 + T^51 + 6*T^50 + T^49 + T^48 + 6*T^47 + 4*T^46 + 2*T^45 + 4*T^44 + 5*T^43 + 5*T^42 + T^40 + 6*T^39 + 3*T^38 + 3*T^37 +3*T^36 + 3*T^35 + 2*T^34 + 2*T^33 + 2*T^32 + 6*T^29 + 4*T^28 + 6*T^27 + 2*T^26 + 2*T^25 + 5*T^24 + 2*T^23 + 5*T^22 + 5*T^21 + 4*T^20 + 4*T^19 + 2*T^18 + 6*T^17 + 2*T^16 + 6*T^15 + 5*T^14 + 2*T^13 + 3*T^12 + 4*T^11 + 4*T^10 + 2*T^9 + 4*T^8 + 4*T^7 + 2*T^6 + T^5 + 6*T^3 + T^2 + 3*T+1] ####################################################### ####################################################### #### #### 3. BEISPIEL ####################################################### ####################################################### k := GF(5^3); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); kx := PolynomialAlgebra(k); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^4+2*y^3*T^3+3*y^2*T^2+T^15+T^11+T^7+2*T^3+1;; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t+Coerce(o,[T+2,T^2+4,11,T+7]); h2 :=t-Coerce(o,[T^2+2*T+34,T,5*T+1,11]); h3 := t+Coerce(o, [T^3+6*T+1,T+1,3*T+46,12*T+6]); h4 := t-Coerce(o,[2+T,T^3+11,2,0]); h5 := t^2+Coerce(o,[T^2+4,2*T+4,11+T,T^2+7]); h6 := t+Coerce(o,[4,T+1,11*T+1,2*T+2]); h7 := t^2+Coerce(o,[1,1,T+1,0]); h8 := t+Coerce(o,[T,1,0,0]); g := h1*h2*h3*h4*h5*h6*h7*h8; Coefgr(g,f,F)[10,81] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); *********************************** [ t + [ 4*T^2 + 3*T + 1, 4*T, 4, 4 ] ], 6, 442, 1 ] Time: 6678.636 s Das Polynom g hat folgende Gestalt: *********************************** t^10 + [ T^3 + 4*T^2 + 1, 4*T^3 + T^2 + T + 1, 4*T, 4 ]*t^9 + [ 3*T^23 + 2*T^22 + 3*T^21 + 4*T^20 + 4*T^19 + T^18 + T^17 + T^16 + T^14 + T^13 + T^12 + 3*T^11 + 3*T^10 + 4*T^9 + 3*T^8 + T^6 + 4*T^5 + 4*T^4 + 4*T^3 + 4*T + 1, T^20 + 4*T^19 + T^18 + 3*T^17 + T^14 + 3*T^13 + T^10 + 3*T^9 + T^8 + 4*T^7 + T^6 + 4*T^5 + 2*T^3 + 3*T^2 + T + 2, 2*T^17 + 3*T^16 + 2*T^15 + 2*T^13 + 3*T^12 + 2*T^11 + 4*T^10 + 3*T^9 + 2*T^8 + 4*T^7 + 3*T^6 + 4*T^3 + 2*T^2 + 2, T^11 + 4*T^10 +T^9 + 3*T^8 + 4*T^5 + 2*T^4 + 4*T^3 + 1 ]*t^8 + [ 4*T^36 + T^35 + 2*T^34 + 2*T^33 + T^31 + 2*T^30 + 3*T^29 + 4*T^28 + 3*T^26 + 3*T^25 + 2*T^23 + 3*T^22 + T^19 + 3*T^18 + 4*T^16 + 3*T^14 + T^13 + 2*T^12 + 4*T^11 + 3*T^10 + T^8 + 3*T^7 + 4*T^6 + 2*T^5 + 2*T^4 + 4*T^3 + 3*T^2 + 4*T + 2, 4*T^33 + T^32 + 2*T^31 + 2*T^30 + 2*T^29 + 2*T^27 + 4*T^26 + 4*T^25 + 3*T^24 + 3*T^22 + T^21 + 4*T^19 + 4*T^18 + 3*T^17 + 3*T^16 + 4*T^15 + T^14 + 2*T^13 + T^12 + 2*T^11 + 4*T^7 + T^6 + 2*T^5 + 3*T^4 + T^3 + 2*T + 1, 2*T^27 + 3*T^26 + T^25 + T^24 + 3*T^22 + 3*T^21 + T^20 + 2*T^19 + 3*T^18 + T^17 + T^16 + 2*T^15 + 3*T^14 + T^12 + T^11 + T^10 + T^9 + 4*T^8 + 2*T^7 + 4*T^6 + 3*T^5 + 4*T^4 + 4*T^3 + 4*T^2 + 1, 2*T^24 + 3*T^23 + T^22 + T^21 + 3*T^20 + 4*T^18 + 2*T^17 + 2*T^16 + 4*T^15 + 4*T^12 + T^11 + 4*T^9 + 3*T^8 + T^7 + 4*T^6 + 2*T^5 + 2*T^4 + 3*T^3 + T^2 + 2*T + 2 ]*t^7 + [ 4*T^48 + 4*T^47 + 4*T^46 + 4*T^45 + 2*T^44 + 2*T^43 + 3*T^42 + 4*T^41 + 2*T^40 + T^39 + T^38 + 3*T^37 + T^36 + T^35 + T^34 + T^33 + T^30 + 4*T^28 +2*T^27 + 2*T^25 + 4*T^24 + T^23 + 3*T^22 + 3*T^20 + T^18 + 3*T^17 + T^16 + T^15 + 2*T^14 + 4*T^13 + 2*T^12 + 3*T^11 + 3*T^10 + T^9 + 4*T^8 + 3*T^7 + 4*T^6 + 2*T^5 + 4*T^3 + T^2 + 2*T + 1, 3*T^42 + 3*T^41 + 3*T^40 + 2*T^39 + 2*T^37 + 4*T^36 + 2*T^35 + 4*T^33 + 3*T^32 + 3*T^31 + 2*T^30 + 4*T^29 + 4*T^28 + 2*T^27 + 3*T^26 + T^24 + 4*T^22 + 4*T^21 + 4*T^20 + 4*T^19 + 3*T^18 + 2*T^17 + 2*T^15 + 2*T^14 + 2*T^13 + 2*T^12 + 3*T^9 + 2*T^8 + 2*T^7 + 3*T^5 + 2*T^4 + 3*T^3 + 4*T^2 + T + 4, 2*T^39 + 2*T^38 + 2*T^37 + 4*T^36 + 3*T^35 + 2*T^34 + 4*T^33 + 2*T^32 + T^31 + 4*T^30 + 2*T^29 + 3*T^28 + 4*T^27 + 2*T^26 + 4*T^25 + 4*T^23 + 4*T^21 + 3*T^20 + T^19 + 2*T^18 + 4*T^16 + T^15 + 3*T^13 + T^11 + 4*T^9 + 4*T^8 + 4*T^7 + 2*T^6 + 3*T^5 + 2*T^4 + 4*T^3 + T^2 + T + 4, 4*T^36 + 4*T^35 + 4*T^34 + T^33 + 3*T^32 + T^31 + T^30 + T^29 + 3*T^28 + 3*T^27 + 3*T^26 + 4*T^25 + 4*T^24 + 3*T^23 + T^22 + 2*T^20 + 3*T^18 + T^17 + 2*T^16 + T^15 + 3*T^14 + 2*T^13 + 2*T^12 + 3*T^11 + 2*T^10 + 4*T^9 + T^8 + 3*T^7 + 4*T^5 + T^4 + 2*T^3 + 4*T^2 + 3*T + 2 ]*t^6 + [ 2*T^54 + 2*T^53 + 2*T^52 + 2*T^49 + 4*T^48 + T^47 + 3*T^46 + 4*T^45 + 4*T^44 + 3*T^43 + 3*T^42 + 2*T^41 + 2*T^40 + 4*T^37 + T^36 + 2*T^35 + 2*T^34 + 2*T^33 + 3*T^31 + 4*T^30 + 4*T^27 + 3*T^26 + 3*T^24 + T^23 + 4*T^21 + 4*T^20 + 2*T^19 + 4*T^18 + 4*T^17 + 3*T^16 + T^15 + T^13 + 2*T^12 + 2*T^11 + 3*T^10 + T^9 + 2*T^6 + 3*T^5 + 3*T^4 + 4*T^3 + 4*T^2 + 3*T + 3, 4*T^51 + 4*T^50 + 4*T^49 + 4*T^48 + T^47 + T^46 + T^44 + 3*T^42 + 2*T^39 + T^38 + 3*T^37 + T^36 + 4*T^35 + 4*T^34 + 3*T^33 + 2*T^32 + 4*T^31 + T^30 + 2*T^28 + 3*T^27 + 2*T^26 + 2*T^25 + 3*T^23 + 3*T^22 + T^21 + 3*T^20 + T^19 + 4*T^18 + 2*T^17 + 2*T^11 + 3*T^10 + T^9 + 4*T^8 + 2*T^7 + T^6 + 2*T^5 + 4*T^4 + T^2 + 3, 2*T^48 + 2*T^47 + 2*T^46 + 4*T^45 + 3*T^44 + 3*T^43 + 3*T^42 + 4*T^39 + T^38 + 4*T^36 + T^35 + 4*T^34 + 3*T^33 + 3*T^32 + 3*T^31 + T^30 + 4*T^29 + 3*T^28 + 3*T^27 + 4*T^26 + 3*T^25 + 3*T^24 + T^23 + T^22 + 2*T^21 + 4*T^20 + 3*T^19 + 3*T^18 + 4*T^17 + 4*T^16 + T^15 + 4*T^14 + 4*T^13 + 3*T^12 + T^11 + 2*T^9 + 2*T^8 + 3*T^7 + 3*T^6 + 2*T^5 + T^4 + T^3 + 4*T + 1, T^42 + T^41 + T^40 + 3*T^38 + 3*T^37 + T^36 + 2*T^35 + T^33 + T^32 + 2*T^31 + 3*T^30 + 4*T^29 + 2*T^27 + 4*T^25 + 4*T^23 + 3*T^22 + 3*T^21 + 2*T^20 + T^19 + 2*T^18 + 2*T^17 + 2*T^16 + 3*T^15 + 3*T^14 + 3*T^12 + 2*T^11 + T^10 + 3*T^9 + 4*T^8 + 3*T^7 + 3*T^4 + 4*T^3 + 2*T^2 + 4*T + 3 ]*t^5 + [ 4*T^57 + T^56 + 4*T^55 + 4*T^54 + T^53 + 3*T^50 + 2*T^49 + 2*T^48 + 4*T^47 + 2*T^46 + 4*T^45 + 2*T^44 + 3*T^43 + 2*T^42 + 3*T^40 + 3*T^39 + 2*T^38 + 4*T^37 + 3*T^36 + T^35 + T^33 + T^31 + 4*T^30 + T^29 + 3*T^28 + 3*T^27 + 2*T^26 + 4*T^23 + 4*T^22 + T^21 + 2*T^19 + 3*T^18 + 3*T^17 + T^15 + 4*T^13 + 3*T^12 + T^11 + 2*T^7 + T^5 + T^4 + 4*T^3 + 3*T^2 + 3*T + 4, T^56 + T^55 + 2*T^53 + 4*T^52 + 4*T^51 + 3*T^50 + T^49 + 2*T^48 + 2*T^47 + 4*T^45 + 3*T^44 + 2*T^43 + 2*T^39 + 2*T^37 + T^36 + 3*T^35 + 3*T^31 + 4*T^30 + T^28 + T^27 + 3*T^24 + 4*T^21 + 2*T^20 + T^19 + T^17 + 2*T^16 + 3*T^15 + T^14 + 3*T^12 + 3*T^10 + 2*T^9 + 4*T^8 + T^6 + T^5 + T^4 + T^2 + 2*T, T^53 + T^52 + 2*T^51 + 4*T^49 + T^48 + T^47 + T^45 + 3*T^44 + 3*T^43 + 4*T^42 + 2*T^41 + T^39 + 4*T^38 + T^37 + 3*T^35 + 4*T^34 + T^33 + 3*T^31 + 2*T^30 + 2*T^29 + T^28 + T^27 + 4*T^26 + T^25 + 3*T^24 + 3*T^23 + 3*T^22 + T^21 + T^20 + 4*T^19 + 2*T^18 + 2*T^17 + 4*T^14 + 3*T^13 + 2*T^12 + 3*T^11 + 2*T^10 + T^9 + 2*T^8 + T^7 + T^6 + 4*T^5 + 3*T^4 + 2*T^3 + 4*T^2 + T + 3, 4*T^50 + 4*T^49 + 4*T^48 + 4*T^47 + 2*T^46 + 2*T^45 + 2*T^44 + 3*T^42 + 3*T^40 + 4*T^39 + 3*T^37 + 3*T^36 + 4*T^35 + 3*T^33 + T^32 + 4*T^31 + 4*T^30 + 4*T^29 + 3*T^27 + 2*T^26 + T^25 + 2*T^23 + 3*T^21 + 4*T^20 + T^19 + 3*T^17 + T^16 + 2*T^15 + 3*T^14 + T^12 + 3*T^10 + 4*T^9 + 2*T^8 + 2*T^6 + 4*T^5 + 2*T^4 + 2*T^3 + T + 3 ]*t^4 + [ 2*T^64 + 3*T^63 + 4*T^61 + 4*T^60 + 3*T^59 + 4*T^58 + 3*T^57 + 2*T^56 + 3*T^55 + 3*T^54 + 3*T^53 + 2*T^52 + T^51 + 3*T^50 + 2*T^49 + 3*T^48 + 2*T^47 + 2*T^46 + 3*T^45+ 2*T^44 + 2*T^43 + 3*T^42 + 4*T^41 + 2*T^40 + 3*T^39 + 2*T^38 + 3*T^37 + T^36 + 4*T^34 + 2*T^32 + T^31 + 4*T^30 + 2*T^29 + 2*T^28 + 2*T^27 + 2*T^26 + T^25+ T^24 + 4*T^22 + 2*T^21 + 4*T^20 + T^18 + 3*T^16 + T^15 + 4*T^11 + 4*T^10 + 3*T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^4 + 4*T^3 + T^2 + 2*T + 2, 3*T^65 + 3*T^64 + 4*T^63 + 4*T^62 + 3*T^61 + 3*T^60 + 4*T^59 + 2*T^58 + 3*T^57 + T^56 + T^55 + 3*T^53 + T^52 + 2*T^51 + T^50 + T^49 + 3*T^48 + 3*T^46 + T^45 + T^44 + T^43 + T^42 + 3*T^41 + T^40 + 4*T^39 + 3*T^38 + 2*T^37 + T^35 + 4*T^34 + 3*T^33 + 3*T^32 + 4*T^31 + 4*T^30 + 3*T^28 + T^27 + 2*T^26 + T^25 + T^24 + T^23 + 3*T^21 + 3*T^20 + 2*T^19 + T^18 + 4*T^17 + T^16 + T^15 + 4*T^14 + 2*T^13 + 4*T^12 + 4*T^9 + 3*T^8 + 4*T^5 + 3*T^4 + 3*T^3 + T^2 + T + 4, 4*T^59 + 4*T^58 + 2*T^57 + 3*T^56 + 4*T^55 + 3*T^54 + 3*T^52 + 4*T^51 + T^50 + 3*T^49 + 4*T^48 + T^47 + T^46 + 3*T^44 + 2*T^43 + 2*T^42 + T^40 + 4*T^39 + 3*T^37 + 4*T^35 + 4*T^34 + 3*T^33 + 4*T^31 + T^30 + 3*T^29 + T^28 + 4*T^27 + T^26 + 2*T^25 + 3*T^24 + 4*T^23 + T^22 + 3*T^21 + 3*T^19 + 2*T^18 + 3*T^17 + 4*T^16 + 4*T^13 + T^12 + 2*T^11 + 4*T^10 + 2*T^9 + 3*T^8 + T^7 + 2*T^6 + T^5 + 4*T^4 + 4*T^2 + 2, 2*T^56 + 2*T^55 + T^54 + 4*T^53 + 4*T^52 + 4*T^51 + 2*T^45 + T^43 + 2*T^42 + T^41 + T^40 + 2*T^39 + 2*T^37 + 2*T^35 + T^34 + 4*T^33 + T^32 + 3*T^30 + T^29 + 4*T^28 + 2*T^26 + 2*T^25 + 2*T^24 + 2*T^23 + 3*T^22 + 4*T^19 + T^18 + 2*T^17 + T^16 + T^15 + 4*T^12 + 4*T^11 + 3*T^10 + 3*T^8 + 3*T^6 + 3*T^5 + 3*T^4 + 2*T^3 + 4*T^2 + 4*T + 3 ]*t^3 + [ 3*T^71 + 3*T^70 + T^69 + 3*T^68 + 3*T^65 + 4*T^64 + 3*T^63 + T^59 + T^58 + 3*T^56 + 3*T^55 + 3*T^54 + 2*T^53 + 3*T^51 + T^50 + 4*T^49 + 3*T^48 + 2*T^46 + 3*T^45 + 2*T^44 + 2*T^43 + 2*T^42 + 3*T^41 + 3*T^40 + T^38 + 3*T^37 + 4*T^36 + 3*T^35 + 2*T^33 + 2*T^32 + 4*T^31 + 4*T^28 + 4*T^27 + T^24 + 3*T^23 + 4*T^22 + T^21 + 4*T^19 + 4*T^18 + 2*T^16 + 4*T^15 + 4*T^14 + 2*T^13 + 4*T^12 + 3*T^11 + T^10 + T^9 + 3*T^8 + 2*T^7 + 4*T^5 + T^4 + 2*T^3 + 4*T^2 + 3*T + 3, 4*T^66 + 4*T^65 + 4*T^64 + 4*T^62 + 3*T^61 + 3*T^58 + 4*T^55 + 2*T^53 + T^52 + T^49 + T^48 + 2*T^47 + 4*T^46 + 3*T^44 + 2*T^43 + 4*T^42 + 4*T^40 + 3*T^39 + 2*T^38 + 4*T^37 + 2*T^36 + 3*T^34 + 2*T^33 + T^32 + 4*T^30 + 3*T^28 + 2*T^26 + 4*T^25 + 3*T^24 + T^23 + 3*T^22 + 3*T^21 + 3*T^20 + 4*T^19 + 4*T^18 + 2*T^17 + 3*T^15 + 4*T^13 + 3*T^12 + 3*T^11 + 2*T^10 + 3*T^9 + 2*T^8 + T^7 + 2*T^6 + T^5 + 4*T^3 + 4*T^2 + 4*T + 4, 3*T^65 + 3*T^64 + 4*T^63 + 4*T^62 + 3*T^61 + 2*T^60 + 3*T^59 + 3*T^58 + 4*T^56 + 2*T^55 + 3*T^54 + 4*T^50 + 2*T^48 + 4*T^47 + T^46 + T^44 + T^43 + 3*T^42 + T^40 + 3*T^39 + 3*T^38 + 2*T^37 + 2*T^35 + 2*T^34 + 3*T^33 + T^32 + 4*T^31 + 3*T^30 + 4*T^29 + 2*T^28 + 2*T^26 + T^24 + T^23 + 2*T^21 + 3*T^20 + 2*T^19 + T^18 + T^17 + 2*T^16 + 2*T^14 + 2*T^12 + 2*T^11 + T^10 + 2*T^9 + 2*T^7 + 3*T^6 + 3*T^5 + 4*T^4 + 3*T^3 + 3*T + 2, 2*T^57 + 2*T^56 + 2*T^55 + 4*T^54 + 3*T^53 + 2*T^51 + T^50 + 4*T^49 + 3*T^48 + 2*T^47 + 4*T^46 + 3*T^45 + 3*T^44 + 3*T^43 + 3*T^42 + 3*T^41 + T^40 + 3*T^38 + T^37 + 3*T^36 + 4*T^35 + 2*T^34 + 3*T^32 + 4*T^31 + 2*T^30 + 3*T^29 + T^28 + 3*T^27 + 3*T^26 + 4*T^24 + 4*T^23 + 2*T^21 + 2*T^19 + T^18 + 3*T^17 + 2*T^16 + 4*T^15 + 3*T^14 + T^13 + T^12 + 2*T^11 + 3*T^10 + 2*T^9 + 3*T^8 + 2*T^7 + T^6 + 3*T^5 + T^4 + T^3 + 4*T^2 + 3*T + 3 ]*t^2 + [ 4*T^71 + 3*T^70 + 2*T^69 + 4*T^68 + T^67 + 4*T^66 + 3*T^63 + 3*T^62 + 3*T^61 + 3*T^60 + 3*T^59 + T^58 + T^57 + T^56 + 2*T^55 + T^53 + T^52 + T^50 + 3*T^49 + 3*T^48 + 3*T^46 + 3*T^45 + 4*T^44 + 3*T^41 + 4*T^40 + 4*T^39 + 2*T^38 + T^37 + 4*T^34 + T^33 + 2*T^32 + 3*T^31 + 4*T^30 + T^29 + 3*T^28 + T^27 + 3*T^25 + 4*T^24 + 3*T^22 + 4*T^21 + 2*T^20 + T^19 + 3*T^18 + T^17 + 3*T^15 + T^11 + 3*T^10 + 4*T^8 + 4*T^7 + T^6 + T^5 + T^4 + 3*T^3 + 3*T^2 + 4*T + 1, 3*T^72 + T^71 + 2*T^70 + 3*T^67 + 3*T^66 + 3*T^65 + 3*T^64 + T^62 + 3*T^59 + 4*T^58 + T^57 + 2*T^56 + T^54 + T^53 + 3*T^50 + 2*T^48 + 4*T^47 + 3*T^46 + 2*T^45 + T^44 + 3*T^43 + T^42 + 3*T^41 + 2*T^40 + T^39 + 4*T^38 + 3*T^37 + 3*T^36 + T^35 + 2*T^34 + T^33 + 3*T^32 + T^31 + 4*T^30 + 3*T^29 + T^28 + 4*T^27 + 3*T^26 + 4*T^24 + 4*T^23 + 3*T^22 + T^21 + T^20 + 2*T^19 + T^18 + 3*T^17 + 3*T^16 + 4*T^15 + 3*T^14 + 3*T^13 + 3*T^12 + 2*T^11 + 2*T^10 + T^8 + 3*T^7 + 3*T^6 + 4*T^5 + T^4 + 2*T^3 + T^2, 2*T^65 + 4*T^64 + T^62 + T^60 + 4*T^59 + T^58 + 2*T^57 + 4*T^56 + 2*T^54 + 3*T^52 + 2*T^51 + 4*T^49 + 2*T^48 + T^47 + 2*T^45 + 4*T^44 + T^43 + 2*T^42 + 4*T^41 + 3*T^40 + 2*T^39 + 4*T^38 + 4*T^36 + 2*T^33 + T^31 + 4*T^30 + 3*T^29 + T^28 + 4*T^27 + 4*T^26 + 2*T^25 + T^24 + T^23 + T^21 + 2*T^20 + 2*T^19 + T^18 + 2*T^17 + 3*T^16 + 4*T^15 + T^13 + 3*T^12 + 2*T^11 + 2*T^10 + 3*T^9 + 4*T^8 + 4*T^5 + 3*T^4 + 4*T^3 + 3*T, 3*T^66 + T^65 + 2*T^64 + 3*T^63 + 2*T^62 + T^61 + 2*T^60 + 3*T^59 + T^58 + 3*T^56 + 3*T^55 + 4*T^54 + 3*T^53 + T^52 + 4*T^51 + 4*T^50 + T^48 + 2*T^47 + 3*T^46 + 4*T^45 + 4*T^44 + T^43 + T^42 + T^40 + T^37 + 4*T^36 + T^35 + 2*T^34 + 3*T^32 + 4*T^31 + 3*T^29 + 2*T^28 + T^27 + 3*T^25 + 4*T^23 + 4*T^22 + 3*T^21 + T^20 + 2*T^19 + 3*T^18 + 4*T^17 + 3*T^16 + 3*T^15 + 3*T^14 + T^13 + 4*T^12 + 4*T^11 + T^5 + 2*T^4 + 4*T^3 + 4*T^2 + 3*T + 2 ]*t + [ 2*T^81 + 4*T^80 + 3*T^79 + 2*T^78 + 3*T^76 + T^75 + 4*T^74 + 4*T^73 + 2*T^72 + 3*T^71 + 2*T^70 + T^69 + 4*T^67 + T^66 + 3*T^65 + 3*T^64 + T^63 + 2*T^62 + 3*T^58 + 4*T^57 + 4*T^56 + 4*T^54 + 4*T^53 + T^52 + 4*T^51 + 3*T^50 + T^48 + T^47 + 3*T^45 + T^44 + 2*T^43 + 2*T^42 + 3*T^41 + T^39 + 3*T^38 + 2*T^37 + 2*T^36 + 2*T^35 + 3*T^34 + 3*T^33 + T^32 + T^31 + T^30 + 4*T^29 + 4*T^28 + T^27 + 4*T^25 + 2*T^24 + T^23 + T^22 + T^19 + 2*T^17 + 2*T^15 + 4*T^13 + T^12 + 3*T^11 + 4*T^10 + 4*T^9 + 4*T^8 + 3*T^7 + 3*T^6 + T^4 + 4*T^3 + 2*T, 3*T^73 + T^72 + T^71 + 4*T^70 + T^68 + 4*T^67 + T^66 + 4*T^64 + T^63 + 4*T^62 + T^60 + 4*T^58 + T^57 + 2*T^56 + 3*T^55 + 3*T^54 + 3*T^53 + 3*T^52 + 4*T^51 + 4*T^50 + T^49 + 2*T^48 + 3*T^46 + 3*T^45 + 3*T^44 + 4*T^43 + 2*T^42 + T^41 + 3*T^40 + 2*T^39 + 4*T^38 + 2*T^37 + T^36 + T^35 + T^34 + T^33 + 4*T^32 + 4*T^30 + 2*T^29 + 4*T^28 + 2*T^27 + 4*T^26 + T^25 + T^24 + 4*T^23 + 4*T^21 + 4*T^20 + 3*T^19 + 3*T^18 + T^16 + 4*T^15 + 4*T^13 + 4*T^11 + 2*T^10 + 4*T^9 + 4*T^8 + T^7 + T^6 + 3*T^5 + 4*T^4 + 2*T^3 + T^2 + 2*T + 2, 3*T^72 + T^71 + 2*T^70 + 3*T^69 + 2*T^68 + T^66 + 3*T^65 + 2*T^63 + T^62 + 2*T^61 + 4*T^59 + T^57 + 2*T^56 + 2*T^55 + 2*T^54 + 3*T^53 + 4*T^52 + T^50 + 3*T^48 + 3*T^45 + 2*T^44 + 4*T^43 + 2*T^41 + 2*T^39 + T^38 + T^37 + 3*T^36 + T^35 + T^34 + 3*T^33 + 4*T^32 + 3*T^31 + 2*T^29 + 3*T^27 + 2*T^26 + 3*T^23 + 3*T^22 + 2*T^21 + 2*T^18 + T^17 + 3*T^16 + 3*T^15 + 4*T^13 + 4*T^12 + 4*T^11 + 3*T^10 + 2*T^9 + T^7 + 4*T^6 + 3*T^4 + 4*T^3 + T^2 + T, 4*T^69 + 3*T^68 + 4*T^67 + 4*T^66 + 3*T^65 + T^64 + 2*T^63 + 2*T^62 + 3*T^61 + T^60 + 3*T^57 + 2*T^56 + 3*T^55 + T^53 + 2*T^52 + 2*T^50 + 2*T^49 + T^48 + T^46 + 3*T^45 + 3*T^44 + 2*T^41 + T^40 + 4*T^39 + 3*T^38 + 4*T^37 + T^36 + 3*T^34 + 2*T^33 + T^32 + 3*T^31 + 3*T^30 + 2*T^29 + 4*T^28 + 2*T^27 + 3*T^26 + 4*T^25 + 4*T^23 + 2*T^22 + 4*T^21 + 2*T^20 + 3*T^14 + 4*T^12 + T^11 + 4*T^10 + 4*T^9+ T^7 + 2*T^6 + 4*T^5 + 4*T^2 + 4*T + 1 ] #################################################### #################################################### #### #### 4. BEISPIEL #################################################### #################################################### k := GF(5,7); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^3+3*y^2*T+4*y*T^2+2*T^3+3; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T,1,1]); h2 :=t-Coerce(o,[1,T,0]); h3 :=t+Coerce(o,[T,2,0]); h4 := t-Coerce(o,[T+4,T,2]); h5 := t+Coerce(o,[T^2+1,0,1]); h6 := t^2+Coerce(o,[1,T^2+4,1]); h7 := t+Coerce(o,[1,T+3,T+1]); h8 := t-Coerce(o,[1,T^2+1,0]); h9 := t+Coerce(o,[2+T,T+1,T]); g := h1*h2*h3*h4*h5*h6*h7*h8 *h9 ;; Coefgr(g,f,F); ##[10,21] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ T, 2, 0 ] ], 2, 82, 1 ] Time: 406.261613 s Das Polynom g hat folgende Gestalt: *********************************** t^10 + [ T^2 + 2*T + 3, 4*T^2 + 1, 2*T + 1 ]*t^9 + [ 2*T^5 + 2*T^4 + 3*T^3 + 3*T, 3*T^5 + T^4 + 3*T^3 + 2*T, T^4 + T^3 + 4*T^2 + 2*T + 4 ]*t^8 + [ 2*T^8 + 2*T^7 + 2*T^6 + 4*T^5 + 4*T^4 + 2*T^3 + 4*T^2 + T + 4, 4*T^8 + T^6 + 4*T^5 + 3*T^3 + 3*T^2 + 4*T + 4, T^6 + 3*T^5 + 2*T^3 + 2*T + 1 ]*t^7 + [ T^10 + T^9 + 3*T^8 + T^6 + T^4 + 2*T^3 + 3*T + 1, 4*T^10 + 4*T^9 + T^8 + 3*T^3 + 3*T^2 + 3*T + 2, 2*T^9 + T^8 + 2*T^7 + 4*T^6 + 4*T^5 + T^4 + 2*T^3 + 2*T^2 ]*t^6 + [ 2*T^13 + 4*T^12 + 2*T^11 + T^10 + 3*T^8 + 4*T^6 + T^5 + 3*T^2 + 2*T + 1, 3*T^12 + 3*T^11 + T^10 + 4*T^9 + 4*T^8 + 3*T^7 + 2*T^6 + T^5 + 4*T^4 + 3*T^3 + T^2 + 2*T + 1, 2*T^11 + T^10 + 3*T^9 + 2*T^8 + 3*T^7 + 4*T^6 + T^5 + 2*T^4 + T^3 + 2*T^2 + 2*T + 4 ]*t^5 + [ 3*T^15 + T^14 + 2*T^13 + T^12 + 4*T^11 + T^9 + 2*T^8 + 3*T^7 + 4*T^6 + T^5 + 2*T^4 + 2*T^3 + 4*T^2 + 2*T + 3, T^14 + 2*T^13 + 2*T^12 + T^11 + 3*T^10 + 3*T^9 + 2*T^8 + T^6 + 3*T^5 + 3*T^4 + 3*T^2 + 3*T + 1, 4*T^12 + 3*T^11 + 4*T^10 + T^9 + 3*T^8 + 3*T^7 + 2*T^6 + 4*T^5 + 4*T^4 + 2*T^3 + 3*T^2 + 2*T + 2 ]*t^4 + [ 2*T^16 + 3*T^14 + 2*T^13 + 2*T^11 + T^10 + T^9 + 3*T^8 + 3*T^7 + T^6 + T^5 + 3*T^4 + 4*T^3 + T^2 + 2*T + 1, 3*T^16 + 3*T^13 + 2*T^12 + 2*T^11 + T^10 + 3*T^8 + T^7 + T^6 + 3*T^5 + 4*T^4 + 4*T^2 + 2*T, T^15 + 4*T^12 + T^11 + T^10 + 2*T^9 + 3*T^8 + 4*T^7 + 4*T^6 + 3*T^4 + T^3 + T^2 + 4*T + 1 ]*t^3 + [ T^18 + 2*T^16 + 2*T^15 + 4*T^14 + 2*T^13 + T^12 + T^11 + 3*T^10 + 3*T^9 + 3*T^8 + 3*T^7 + 3*T^6 + 4*T^5 + 3*T^4 + T^3 + 2*T + 3, 3*T^17 + 4*T^16 + 4*T^15 + 4*T^13 + T^11 + T^8 + T^7 + T^6 + 2*T^5 + 2*T^3 + 3*T^2, 2*T^16 + 3*T^14 + 3*T^12 + 2*T^10 + T^9 + 2*T^8 + 2*T^7 + 3*T^6 + T^5 + T^4 + 2*T^3 + 3*T^2 + 1 ]*t^2 + [ 3*T^20 + 4*T^19 + 3*T^18 + 4*T^17 + 4*T^16 + 2*T^15 + 2*T^13 + T^11 + T^10 + 3*T^9 + 3*T^8 + 4*T^7 + T^6 + 3*T^5 + 4*T^4 + 3*T^3 + T^2 + 4*T + 1, T^19 + 2*T^18 + 3*T^17 + 3*T^15 + 4*T^14 + 4*T^13 + 4*T^12 + 2*T^11 + 2*T^10 + 4*T^8 + 2*T^7 + 3*T^6 + 3*T^5 + 2*T^3 + 4*T^2 + T + 3, 3*T^17 + 4*T^16 + 3*T^14 + 3*T^13 + 3*T^12 + 2*T^11 + 3*T^10 + 3*T^9 + 2*T^8 + 4*T^7 + 4*T^6 + 2*T^5 + 2*T^4 + 4*T^3 + 3*T^2 + 2*T ]*t + [ 3*T^21 + T^19 + 2*T^18 + 2*T^17 + T^16 + 4*T^15 + T^14 + 3*T^13 + T^12 + 3*T^11 + 2*T^10 + 2*T^9 + T^8 + 2*T^7 + 4*T^3 + 4*T^2 + 2*T + 2, 2*T^20 + 2*T^19 + 4*T^18 + 2*T^17 + 2*T^16 + 4*T^15 + 4*T^14 + 2*T^12 + 3*T^11 + 2*T^10 + 4*T^9 + 2*T^8 + T^7 + 2*T^6 + 3*T^4 + 2*T^3 + T^2 + 3, 2*T^19 + 4*T^18 + 4*T^17 + T^16 + 3*T^15 + 4*T^14 + 4*T^13 + 3*T^12 + 4*T^11 + 2*T^10 + 2*T^9 + T^7 + 4*T^6 + 3*T^5 + 3*T^4 + 3*T^3 + 3*T^2 + 3*T + 1 ] ####################################################### ####################################################### #### #### 5. BEISPIEL ####################################################### ####################################################### k := GF(7,5); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^5+2*y^2*T+2*y*T+T+1;; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T+1,2*T,T,0,0]); h2 :=t-Coerce(o,[1,T,T,0,T]); h3 :=t+Coerce(o,[T^2+1,1,T^3,1,0]); h4 := t^3+Coerce(o,[3,6,T+1,T+1,1]); h5 := t^2+Coerce(o,[T^2+1,2*T+1,4*T,1,6+T]); h6 := t+Coerce(o,[T+1,4*T^2+1,4*T^3,1,1]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F); [9,15] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); *********************************** [ [ t + [ 6, 6*T, 6*T, 0, 6*T ] ], 4, 294, 1 ] Time: 351.694991 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ T^2 + 2*T + 2, 4*T^2 + T + 2, 5*T^3, 2, 6*T + 1 ]*t^8 + [ 6*T^4 + T^3 + 4*T^2 + 2*T, 5*T^5 + 6*T^4 + 5*T^3 + T^2 + T + 2, 2*T^4 + 4*T^3 + 5*T^2 + T, 3*T^4 + 3*T^2 + 3, 4*T^6 + 6*T^3 + T^2 + 6*T + 2 ]*t^7 + [ 6*T^9 + 2*T^8 + 5*T^6 + 2*T^5 + 2*T^4 + 2*T^3 + 4*T^2 + 2*T + 4, 5*T^9 + 6*T^8 + 5*T^7 + 5*T^6 + 6*T^5 + 3*T^4 + 3*T^3 + 2*T^2 + 5*T + 6, 5*T^9 + 6*T^8 + 6*T^7 + 2*T^6 + 2*T^5 + 5*T^4 + 4*T^3 + 6*T^2 + 4, 4*T^8 + 6*T^7 + 6*T^6 + 5*T^5 + 2*T^3 + 3*T^2 + T + 4, T^8 + 5*T^7 + 2*T^6 + 6*T^5 + 6*T^4 + 2*T^2 + 4*T + 3 ]*t^6 + [ 6*T^10 + 2*T^9 + T^8 + 6*T^7 + 5*T^6 + 6*T^5 + 4*T^4 + 3*T^3 + 5*T^2 + 5*T, 2*T^10 + 2*T^8 + 5*T^7 + 5*T^6 + T^5 + 5*T^3 + 4*T^2 + 3*T + 5, 5*T^10 + 4*T^9 + 5*T^7 + T^6 + 5*T^5 + 5*T^4 + 2*T^3 + T^2 + 5*T + 4, 6*T^10 + 5*T^9 + 2*T^5 + 3*T^3 + 6*T + 4, 5*T^10 + 6*T^9 + 2*T^8 + 6*T^7 + 3*T^6 + 5*T^4 + 6*T^3 + T^2 + 3*T + 2 ]*t^5 + [ 3*T^11 + 4*T^10 + T^8 + T^7 + 5*T^6 + 3*T^5 + 6*T^4 + 5*T^2 + T + 1, T^11 + 4*T^10 + 4*T^9 + 6*T^8 + 3*T^7 + 5*T^6 + 3*T^5 + 6*T^4 + 6*T^3 +4*T^2 + 6, 3*T^11 + T^10 + T^9 + T^8 + 4*T^7 + 2*T^5 + 4*T^4 + 3*T^3 + 2*T^2 + 2*T + 4, 4*T^11 + T^10 + T^9 + 3*T^8 + T^7 + 3*T^5 + T^4 + T^3 + 3*T^2 + 4, 3*T^10 + 3*T^9 + T^7 + T^6 + 6*T^5 + 2*T^4 + 5*T^3 + T + 2 ]*t^4 + [ 3*T^13 + 3*T^12 + T^11 + 4*T^10 + 6*T^8 + 6*T^7 + 2*T^5 + T^4 + 6*T^3 + 5*T + 1, 6*T^13 + 5*T^11 + 3*T^10 + 2*T^9 + 6*T^8 + 5*T^7 + 5*T^6 + T^5 + 3*T^3 + 2*T^2 + 2*T+ 6, 6*T^13 + 6*T^11 + 2*T^10 + 6*T^9 + 6*T^8 + 5*T^7 + 3*T^6 + T^5 + T^4 + T^3 + 4*T^2 + 3*T + 5, 2*T^12 + 4*T^11 + 5*T^10 + 4*T^9 + 5*T^6 + 6*T^4 + 6*T, 4*T^12 + 6*T^11 + T^10 + 2*T^9 + 5*T^8 + 6*T^7 + 6*T^6 + 2*T^5 + 2*T^4 + 6*T^3 + 2*T^2 + T + 1 ]*t^3 + [ 6*T^12 + T^11 + 4*T^8 + 6*T^7 + 2*T^6 + 6*T^4 + 6*T^3 + 6*T^2 + 5*T + 3, T^12 + T^11 + T^9 + 6*T^8 + 5*T^7 + 5*T^6 + 5*T^5 + 5*T^4 + 3*T^3 + 5*T^2 + 3*T + 2, 6*T^12 + 4*T^11 + T^10 + 6*T^9 + 4*T^8 + 6*T^7 + 4*T^6 + 5*T^5 + 3*T^4 + 2*T^3 + 3*T^2 + 1, 3*T^12 + 6*T^11 + T^10 + 4*T^9 + 2*T^8 + 3*T^7 + 5*T^6 + 2*T^5 + 6*T^4 + T^3 + 5*T^2 + 4*T + 3, 4*T^12 + 5*T^11 + 2*T^10 + T^9 + 3*T^8 + 2*T^5 + 5*T^4 + T^3 + 6*T + 4 ]*t^2 + [ 3*T^12 + 6*T^11 + 3*T^10 + 3*T^9 + 4*T^8 + T^7 + 4*T^6 + 5*T^5 + 2*T^4 + 2*T^3 + 4*T + 5, 3*T^13 + 5*T^12 + 3*T^11 + 3*T^10 + 3*T^8 + 4*T^7 + 2*T^6 + 6*T^5 + 2*T^4 + 2*T^3 + T + 1, 6*T^13 + T^12 + 6*T^11 + 5*T^10 + 6*T^9 + 6*T^8 + 6*T^7 + 4*T^6 + T^5 + 3*T^4 + 6*T^3 + 4*T + 5, 6*T^13 + 3*T^12 + 4*T^10 + T^9 + 4*T^8 + T^7 + 5*T^6 + 2*T^4 + 2*T^2, 4*T^12 + 2*T^11 + 6*T^10 + 5*T^9 + 3*T^8 + 2*T^7 + 3*T^6 + T^5 + 2*T^4 + 4*T^3 + 4*T^2 + T + 2 ]*t + [ T^15 + 2*T^14 + 4*T^13 + 3*T^12 + 5*T^11 + 6*T^10 + 3*T^9 + 5*T^8 + 5*T^5 + 5*T^3 + 2*T^2 + 6*T + 4, 2*T^15 + 4*T^14 + 3*T^13 + 2*T^12 + 5*T^11 + 4*T^10 + T^9 + 5*T^8 + 3*T^7 + T^6 + T^5 + 4*T^4 + 4*T^3 + 3*T^2 + 5, 2*T^15 + 5*T^14 + T^12 + 2*T^11 + 3*T^10 + 5*T^8 + 5*T^6 + 5*T^5 + 6*T^4 + 6*T^3 + 3*T^2 + 2*T + 5, 5*T^14 + 2*T^13 + 5*T^11 + 5*T^10 + T^9 + 3*T^8 + 3*T^7 + T^5 + 3*T^4 + 2*T^3 + T^2 + 2*T + 1, 4*T^14 + 4*T^13 + 5*T^12 + 3*T^11 + 2*T^10 + T^9 + 4*T^8 + 6*T^7 + 3*T^5 + 5*T^4 + 5*T^3+ 3*T^2 + 4 ] ####################################################### ####################################################### #### #### 6. BEISPIEL ##################################################### ##################################################### k := GF(11^2); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); kx := PolynomialAlgebra(k); AssignNames_ (kTy, ["y"]);Coefgr(g,f,F); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^7+2*y^3*T^3+3*y*T+T+1; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[5*T+2,T,1,0,0,0,0]); h2 :=t-Coerce(o,[T,1,0,0,0,0,0]); h3 :=t^2+Coerce(o,[T+1,7,0,0,0,0,0]); h4 :=t+Coerce(o,[T+1,T^2,0,0,0,0,0]); h5 :=t-Coerce(o,[T+1,2*T+1,0,0,0,0,0]); h6 :=t+Coerce(o,[T+1,T,0,0,0,0,0]); h7 :=t+Coerce(o,[3*T^3+T^2+T+1,1,0,0,0,0,0]); g := h1*h2*h3*h4*h5*h6*h7;; Coefgr(g,f,F);[9,10]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ t + [ 10*T, 10, 0, 0, 0, 0, 0 ] ], 3, 168, 1 ]--->>>Time: 261.785302 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ 3*T^3 + T^2 + T + 2, T^2 + 10*T + 10, 0, 0, 0, 0, 0 ]*t^8 + [ 3*T^3 + 10*T^2 + 4*T + 3, 3*T^5 + 9*T^4 + 4*T^3 + 6*T^2 + 3*T + 3, 10*T^3 + 8*T^2 + 10*T, 0, 0, 0, 0 ]*t^7 + [ 5*T^5 + 7*T^4 + 7*T^3 + 2*T^2 + 9*T + 4, 8*T^6 + T^5 + 2*T^4 + 8*T^3 + T^2 + 9*T + 7, 8*T^6 + 9*T^5 + 4*T^4 + 9*T^3 + 2*T^2 + 4*T + 6, 9*T^4 + 10*T^3, 0, 0, 0 ]*t^6 + [ 8*T^5 + T^4 + 10*T + 1, 8*T^7 + 3*T^6 + 4*T^5 + 9*T^4 + 8*T^3 + 5*T^2 + 5*T + 2, 5*T^7 + 3*T^6 + 9*T^5 + T^4 + 7*T + 6, 5*T^7 + 9*T^6 + T^5 + 3*T^4 + T^3 + 4*T^2 + T + 9, 10, 0, 0 ]*t^5 + [ 3*T^7 + 7*T^6 + 10*T^5 + 10*T^4 + 3*T^3 + 10*T^2 + 9*T + 9, 3*T^8 + 9*T^7 + 2*T^6 + 8*T^5 + 7*T^4 + 5*T^3 + T^2 + 9*T + 1, 9*T^8 + 3*T^7 + 7*T^6 + 10*T^5 + 8*T^4 + 2*T^3 + T^2 + 8*T + 4, 6*T^8 + 9*T^6 + 6*T^5 + 6*T^3 + 2, 6*T^7 + 2*T^6 + T^5 + 7*T^4 + 4*T^3 + 2*T, 10*T^2 + T + 1, 0 ]*t^4 + [ 4*T^6 + 3*T^5 + 2*T^4 + 7*T^3 + 2*T^2 + 6*T + 8, 4*T^8 + 4*T^7 + 5*T^6 + 3*T^5 + 6*T^4 + 2*T^3 + 9*T^2 + 2*T + 7, 5*T^8 + 4*T^7 + 3*T^6 + 9*T^5 + 6*T^4 + 3*T^3 + 2*T^2 + 6*T + 5, 2*T^8 + 2*T^7 + 3*T^6 + 3*T^5 + 10*T^4 + T^3 + T^2 + 9*T + 9, 5*T^8 + 5*T^7 + 6*T^6 + 3*T^5 + 6*T^4 + 4*T + 10, 5*T^7 + 9*T^6 + 9*T^5 + 7*T^4 + 10*T^3 + 10*T^2 + 7*T + 6, T^3 + 3*T^2 + T ]*t^3 + [ 7*T^8 + 6*T^7 + T^5 + 4*T^3 + 7*T^2 + 10*T + 7, 7*T^9 + 5*T^8 + 2*T^7 + 9*T^6 + 5*T^5 + 5*T^4 + 2*T^3 + 8*T^2 + 8*T, 2*T^9 + 3*T^8 + 7*T^7 + 2*T^6 + 7*T^5 + 8*T^4 + 6*T^3 + 10*T^2 + 2*T + 10, T^9 + 2*T^8 + 5*T^7 + 5*T^6 + 9*T^5 + 4*T^4 + 8*T^3 + 4*T^2 + 2*T + 2, 6*T^9 + 2*T^8 + 5*T^7 + 4*T^6 + 6*T^5 + 4*T^4 + 10*T^3 + 3*T^2 + 4*T + 6, T^8 + 6*T^7 + 10*T^6 + 7*T^5 + 3*T^3 + 4*T^2 + 2, 6*T^7 + 5*T^6 + 7*T^5 + 8*T^4 + 7*T^3 + 8*T^2 + 8*T + 7 ]*t^2 + [ 7*T^7 + 4*T^6 + 3*T^5 + 10*T^4 + 4*T^3 + 6*T^2 + 4*T + 9, 7*T^9 + 3*T^8 + 9*T^7 + 10*T^6 + 7*T^5 + 5*T^4 + 8*T^3 + 8*T^2 + 4*T + 8, T^7 + 10*T^6 + 3*T^5 + 3*T^4 + 4*T^3 + 4*T^2 + 3*T + 5, 8*T^9 + 5*T^8 + 4*T^6+ 3*T^5 + 6*T^4 + 5*T^3 + 2*T^2 + 2*T + 6, 5*T^9 + 9*T^8 + 2*T^7 + 7*T^6 + T^5 + 7*T^4 + 6*T^3 + 4*T^2 + 5*T + 9, 7*T^8 + 2*T^7 + 4*T^6 + 10*T^5 + 2*T^3 + 2*T^2 + 4*T + 1, 2*T^7 + 8*T^6 + 8*T^5 + 6*T^4 + 8*T^3 + 8*T^2 + 6*T + 3 ]*t + [ 4*T^9 + 7*T^8 + 9*T^7 + 9*T^6 + 10*T^5 + 2*T^4 + 2*T^3 + 2*T^2 + 6*T, 4*T^10 + 5*T^9 + 9*T^7 + 2*T^5 + 3*T^4 + 7*T^3 + 7*T^2 + 2, 4*T^10 + 10*T^9 + T^8 + 8*T^7 + 3*T^6 + 4*T^5 + 4*T^4 + 4*T^3 + 7*T^2 + T + 7, 10*T^10 + 7*T^9 + 10*T^8 + 7*T^7 + T^6 + 8*T^5 + 3*T^4 + 6*T^3 + 4*T^2 + 10*T + 9, 6*T^10 + 2*T^9 + 10*T^8 + T^7 + 7*T^6 + T^5 + 7*T^4 + 2*T^3 + 5*T^2 + 3*T + 1, 10*T^9 + 6*T^8 + 5*T^7 + 7*T^6 + 5*T^5 + 6*T^4 + 6*T^3 + 7*T^2 + 10*T + 4, 2*T^8 + T^7 + 6*T^6 + 4*T^5 + 9*T^4 + 6*T^3 + 10*T + 7 ] ####################################################### ####################################################### #### #### 7. BEISPIEL ######################################################### ######################################################### k := GF(13^2); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); kx := PolynomialAlgebra(k); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^5+2*y^3*T+3*y*T+T+1; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; h1 := t+Coerce(o,[3*T^2+2,T,1,1,2]); h2 :=t-Coerce(o,[4*T^4,T^2+1,T^5,1,4]); h3 :=t^2+Coerce(o,[T+1,1,0,0,0]); h4 :=t+Coerce(o,[3*T^11+T^10+3*T^5+2*T^7+T+1,T,1,T^11,T]); h5 :=t^2+Coerce(o,[T^2+1,T^2+1+T,1,1,0]); h6 :=t^2+Coerce(o,[T^11+3*T^3+1,2*T^2+1+T,T,3+T^2,0]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F);[9,33] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ************************************* [ [ ot.1 + [ 9*T^4, 12*T^2 + 12, 12*T^5, 12, 9 ] ], 5, 1500,1 ] Time: 1390.066509 s Das Polynom g hat folgende Gestalt: *********************************** ot.1^9 + [ 3*T^11 + T^10 + 2*T^7 + 3*T^5 + 9*T^4 + 3*T^2 + T + 3, 12*T^2 + 2*T + 12, 12*T^5 + 2, T^11, T + 11 ]*ot.1^8 + [ T^17 + T^16 + T^15 + 9*T^14 + 5*T^13 + 11*T^12 + 11*T^11 + 2*T^10 + 7*T^9 + 2*T^6 + 3*T^5 + 2*T^4 + 8*T^3 + T + 8, 3*T^17 + 11*T^13 + 12*T^12 + 11*T^11 + 12*T^10 + 11*T^9 + 2*T^8 + 9*T^7 + 9*T^6 + 4*T^5 + T^3 + T^2 + 3*T + 7, 10*T^16 + 12*T^15 + 5*T^11 + 11*T^10 + 2*T^7 + 5*T^6 + 5*T^4 + 12*T^3 + 5*T^2 + T + 10, 2*T^17 + 9*T^15 + 8*T^13 + 4*T^12 + 2*T^11 + 11*T^4 + 2*T^3 + T^2 + 10*T + 8, 12*T^13 + T^12 + 6*T^11 + 11*T^10 + 11*T^7 + 4*T^6 + T^5 + 5*T^4 + 8*T^3 + 11*T^2 + 11*T + 3 ]*ot.1^7 + [ 3*T^22 + T^21 + 11*T^19 + 5*T^18 + 10*T^17 + 11*T^16 + 5*T^15 + 6*T^14 + 7*T^13 + 5*T^12 + 6*T^10 + 7*T^9 + T^8 + 12*T^7 + 4*T^6 + 12*T^5 + 5*T^3 + 9*T^2 + 8*T + 5, 7*T^19 + T^18 + 11*T^17 + 2*T^16 + 8*T^15 + 8*T^14 + 6*T^13 + 11*T^12 + 4*T^11 + 12*T^10 + 2*T^9 + 10*T^8 + 5*T^7 + 9*T^6 + 6*T^5 + T^4 + 5*T^3 + 6*T^2 + 9*T + 7, 10*T^18 + 12*T^17 + 7*T^16 + 8*T^15 + 11*T^14 + 4*T^13 + 3*T^12 + 11*T^11 + 12*T^10 + 3*T^9 + 8*T^8 + 4*T^7 + 7*T^6 + 10*T^5 + 5*T^4 + 6*T^3 + 10*T^2 + T + 10, T^22 + 9*T^19 + 11*T^18 + 6*T^17 + 8*T^16 + 10*T^15 + 12*T^14 + T^13 + 8*T^12 + 8*T^11 + 4*T^10 + 6*T^9 + 11*T^8 + 2*T^7 + 5*T^6 + 8*T^5 + 7*T^4 + 12*T^3 + 6*T^2 + 4*T + 12, 11*T^18 + 4*T^17 + 7*T^16 + 3*T^15 + 10*T^14 + T^13 + 9*T^12 + 6*T^11 + T^10 + 12*T^9 + 12*T^8 + 12*T^6 + 8*T^5 + 8*T^4 + 7*T^3 + 3*T^2 + 7*T + 2 ]*ot.1^6 + [ T^28 + T^27 + T^26 + 9*T^25 + 5*T^24 + 11*T^23 + 10*T^22 + 2*T^21 + 10*T^20 + 5*T^19 + 5*T^18 + 8*T^17 + 6*T^16 + 8*T^15 + 4*T^14 + 8*T^13 + 11*T^12 + 4*T^11 + 9*T^10 + 12*T^9 + 10*T^7 + 11*T^6 + 2*T^5 + T^4 + 11*T^3 + 10*T^2 + 3*T + 5, 3*T^28 + 11*T^24 + 12*T^23 + 11*T^22 + 12*T^21 + 5*T^20 + 9*T^19 + 10*T^18 + 12*T^17 + 12*T^16 + 8*T^15 + 11*T^13 + T^12 + 4*T^10 + 12*T^8 + 10*T^7 + 8*T^6 + 3*T^5 + 3*T^4 + 3*T^3 + 6*T^2 + 12*T + 11, 10*T^27 + 12*T^26 + 5*T^22 + 11*T^21 + 7*T^20 + 4*T^18 + 12*T^17 + 4*T^16 + 10*T^15 + 7*T^14 + 11*T^13 + 4*T^12 + 11*T^11 + 5*T^10 + T^9 + 3*T^8 + 10*T^7 + 7*T^5 + 8*T^4 + T^3 + 7*T^2 + T + 11, 2*T^28 + 9*T^26 + 8*T^24 + 4*T^23 + 2*T^22 + 6*T^20 + 5*T^19 + 9*T^18 + 3*T^17 + 12*T^16 + 12*T^15 + 12*T^14 + 6*T^13 + 12*T^12 + 6*T^11 + 3*T^10 + 7*T^9 + 7*T^8 + 12*T^7 + 12*T^6 + 7*T^5 + 7*T^4 + 3*T^3 + 2*T^2 + 2*T + 8, 12*T^24 + T^23 + 6*T^22 + 11*T^21 + 9*T^20 + 9*T^19 + 7*T^18 + 7*T^17 + 2*T^16 + 10*T^15 + 11*T^14 + 4*T^13 + 10*T^12 + 9*T^11 + 10*T^10 + 2*T^9 + 7*T^8 + 3*T^7 + 9*T^6 + 11*T^5 + 7*T^4 + 8*T^3 + 5*T^2 + 9*T + 2 ]*ot.1^5 + [ 11*T^30 + 3*T^29 + 10*T^28 + 8*T^27 + 9*T^26 + 10*T^25 + 2*T^24 + 5*T^23 + 10*T^22 + 10*T^21 + 4*T^20 + 8*T^19 + 3*T^16 + 3*T^15 + 8*T^14 + 5*T^12 + 7*T^11 + 7*T^10 + 7*T^9 + 12*T^8 + 12*T^7 + T^6 + 10*T^5 + 5*T^4 + 7*T^3 + 5*T^2 + 7*T + 5, 7*T^30 + T^29 + 11*T^28 + 2*T^27 + 8*T^26 + 9*T^25 + 5*T^24 + 7*T^23 + 4*T^22 + 6*T^21 + 10*T^20 + 12*T^19 + 7*T^18 + 10*T^17 + 5*T^16 + 8*T^15 + 6*T^14 + 2*T^13 + 8*T^12 + 4*T^11 + 12*T^10 + 12*T^9 + 10*T^8 + 5*T^6 + 6*T^5 + 3*T^4 + 4*T^3 + 12*T + 4, 10*T^29 + 12*T^28 + 8*T^27 + 8*T^26 + T^25 + 4*T^24 + 9*T^23 + 7*T^22 + 6*T^21 + 7*T^20 + 12*T^19 + 8*T^18 + 4*T^17 + T^16 + 7*T^15 + 6*T^14 + 11*T^13 + 6*T^12 + 4*T^11 + 9*T^10 + 8*T^9 + 6*T^8 + 10*T^7 + T^6 + 2*T^5 + 3*T^3 + 8*T^2 + 10*T + 1, 9*T^30 + 11*T^29 + 6*T^28 + 8*T^27 + 10*T^26 + 9*T^25 + 7*T^23 + 4*T^22 + 11*T^21 + 12*T^20 + 7*T^19 + 10*T^18 + 5*T^17 + 8*T^16 + 5*T^14 + 4*T^13 + 2*T^12 + 10*T^11 + 8*T^10 + 10*T^9 + 8*T^8 + 8*T^7 + T^6 + 5*T^4 + 12*T^3 + 9*T^2 + 11, 11*T^29 + 4*T^28 + 7*T^27 + 3*T^26 + 12*T^25 + 12*T^24 + 2*T^22 + 7*T^21 + 7*T^20 + 4*T^19 + 8*T^18 + T^17 + 9*T^16 + 10*T^15 + 7*T^14 + 7*T^13 + 4*T^12 + 11*T^11 + 3*T^10 + 9*T^9 + 4*T^8 + 6*T^7 + 2*T^6 + 8*T^5 + 5*T^4 + 10*T^3 + 12*T^2 + 2*T + 5 ]*ot.1^4 + [ T^30 + 5*T^28 + 8*T^27 + 7*T^26 + 8*T^25 + 9*T^24 + 6*T^23 + 10*T^22 + 6*T^21 + 12*T^20 + 7*T^19 + 2*T^18 + 2*T^17 + 4*T^16 + 6*T^15 + 12*T^14 + 4*T^13 + 9*T^12 + 8*T^11 + 8*T^10 + 10*T^9 + T^7 + 4*T^6 + 5*T^5 + 7*T^4 + 9*T^3 + 11*T^2 + T + 12, 4*T^30 + 10*T^29 + 4*T^28 + 8*T^27 + 12*T^26 + 4*T^25 + 5*T^24 + 4*T^23 + 11*T^22 + 5*T^21 + 3*T^20 + 6*T^19 + 5*T^18 + 9*T^17 + 11*T^16 + 4*T^15 + 2*T^13 + 10*T^12 + 3*T^11 + 10*T^10 + 2*T^9 + 4*T^8 + 4*T^7 + 12*T^6 + 7*T^5 + 10*T^4 + 7*T^2 + 2, 3*T^30 + 7*T^29 + 3*T^28 + 6*T^27 + 10*T^26 + 12*T^25 + T^24 + 4*T^23 + T^22 + 10*T^21 + 3*T^20 + 3*T^19 + 6*T^18 + 10*T^17 + 8*T^16 + 4*T^15 + 12*T^14 + 9*T^13 + T^12 + 3*T^11 + 10*T^8 + 12*T^7 + 9*T^6 + T^5 + 2*T^4 + 5*T^3 + 9*T^2 + 7*T + 2, 2*T^30 + 8*T^29 + 6*T^28 + 2*T^27 + 8*T^26 + 2*T^25 + 5*T^24 + 10*T^23 + 3*T^22 + 8*T^21 + 11*T^20 + T^19 + 7*T^18 + 11*T^17 + 11*T^16 + 11*T^15 + 5*T^14 + 4*T^13 + 11*T^12 + 12*T^11 + 2*T^10 + T^9 + 3*T^8 + 8*T^7 + T^6 + 6*T^5 + 12*T^4 + 3*T^3 + 6*T + 2, 2*T^30 + 11*T^29 + 3*T^28 + T^27 + 11*T^26 + 11*T^25 + 2*T^24 + 12*T^23 + 10*T^22 + T^21 + 5*T^20 + 2*T^19 + 6*T^18 + 4*T^17 + 4*T^16 + 9*T^15 + T^14 + 10*T^13 + 8*T^12 + 12*T^11 + 2*T^10 + 12*T^9 + 9*T^8 + T^6 + 8*T^5 + 3*T^4 + 3*T^3 + 11*T^2 + 2*T + 3 ]*ot.1^3 + [ T^31 + 8*T^29 + 2*T^28 + 12*T^27 + 6*T^26 + T^25 + 2*T^24 + 12*T^23 + 10*T^22 + 7*T^21 + 8*T^19 + 8*T^18 + 6*T^17 + 3*T^16 + 7*T^15 + 9*T^14 + 9*T^13 + 6*T^12 + 4*T^11 + 5*T^10 + 11*T^9 + 2*T^8 + 10*T^7 + 4*T^6 + 9*T^5 + 8*T^4 + 8*T^3 + 4*T^2 + 4*T + 1, 11*T^32 + 7*T^31 + 9*T^30 + 6*T^29 + 5*T^28 + 11*T^27 + 10*T^25 + 8*T^24 + 3*T^23 + 2*T^21 + 6*T^20 + 6*T^19 + 12*T^18 + T^17 + 11*T^16 + 4*T^15 + T^14 + T^13 + 9*T^12 + 6*T^11 + 3*T^10 + 11*T^9 + 6*T^7 + 5*T^6 + 9*T^5 + 7*T^4 + 9*T^2 + 6*T + 7, 7*T^32 + 4*T^31 + 8*T^30 + 5*T^27 + 5*T^26 + 2*T^25 + 6*T^24 + 9*T^23 + 8*T^22 + 2*T^19 + 10*T^18 + 9*T^17 + 8*T^15 + 6*T^14 + 6*T^13 + 3*T^11 + 11*T^10 + 8*T^9 + 2*T^8 + 2*T^7 + 9*T^6 + 4*T^5 + 6*T^4 + 3*T^3 + 8*T^2 + 4*T, 2*T^30 + 11*T^29 + 5*T^28 + 2*T^27 + 8*T^26 + 6*T^25 + T^24 + 7*T^23 + 12*T^22 + 9*T^21 + 2*T^20 + 3*T^19 + T^18 + T^17 + 11*T^16 + 5*T^14 + 9*T^12 + T^11 + 10*T^10 + 7*T^9 + 6*T^8 + 12*T^7 + 5*T^6 + 8*T^5 + 10*T^4 + 4*T^3 + 5*T^2 + T + 6, 9*T^32 + 8*T^29 + 2*T^28 + 9*T^27 + 5*T^26 + 4*T^24 + 5*T^23 + 12*T^22 + 7*T^21 + 2*T^20 + 11*T^19 + 2*T^18 + T^17 + 3*T^16 + 4*T^15 + 7*T^14 + 3*T^13 + 8*T^12 + 6*T^11 + 8*T^10 + 7*T^8 + 11*T^7 + 2*T^6 + 5*T^5 + 5*T^4 + 11*T^3 + 2*T^2 + 12 ]*ot.1^2 + [ 12*T^31 + 3*T^29 + 7*T^28 + 2*T^26 + 6*T^25 + 3*T^24 + 2*T^23 + 5*T^22 + 2*T^21 + 2*T^20 + 11*T^19 + 7*T^18 + 11*T^16 + 10*T^15 + T^14 + 11*T^13 + 8*T^12 + 12*T^11 + 8*T^10 + 4*T^9 + T^8 + 11*T^7 + T^6 + 12*T^5 + 5*T^4 + 8*T^3 + 11*T^2 + 6, 11*T^31 + 5*T^30 + 3*T^29 + 7*T^28 + 5*T^27 + 5*T^26 + 11*T^25 + 10*T^24 + 5*T^23 + 5*T^22 + 12*T^21 + 6*T^20 + 8*T^19 + 7*T^18 + 9*T^17 + T^16 + 10*T^15 + 2*T^14 + 5*T^13 + 2*T^12 + 11*T^11 + 2*T^10 + 11*T^9 + 9*T^8 + 12*T^7 + T^6 + 10*T^5 + 10*T^4 + 4*T^3 + 12*T^2 + 12*T + 5, 3*T^31 + T^30 + 4*T^29 + 2*T^27 + 8*T^26 + 9*T^25 + 9*T^24 + T^23 + 4*T^22 + 10*T^21 + 5*T^19 + 5*T^18 + 2*T^17 + T^16 + 9*T^14 + 5*T^12 + 10*T^11 + 9*T^10 + 11*T^9 + 8*T^8 + 11*T^7 + T^5 + 8*T^4 + 2*T^3 + T + 12, 11*T^31 + 2*T^30 + 2*T^29 + T^28 + 2*T^27 + 5*T^26 + 11*T^25 + 12*T^24 + 12*T^21 + 4*T^20 + 11*T^19 + 5*T^18 + 9*T^17 + 7*T^16 + 7*T^15 + 9*T^14 + 6*T^12 + 6*T^11 + T^10 + 4*T^9 + 9*T^8 + 2*T^7 + 5*T^5 + 6*T^4 + 6*T^3 + 12*T^2 + 8*T, 2*T^31 + 2*T^30 + 5*T^29 + 6*T^28 + 12*T^27 + T^26 + 11*T^25 + 3*T^24 + 8*T^23 + 3*T^22 + 10*T^21 + 12*T^20 + 10*T^19 + T^18 + 8*T^16 + T^15 + 2*T^14 + 8*T^13 + 7*T^12 + 11*T^11 + 2*T^10 + 9*T^9 + 10*T^8 + 7*T^7 + 7*T^6 + 5*T^5 + 6*T^4 + 7*T^2 + 3*T ]*ot.1 + [ 4*T^33 + 7*T^32 + 7*T^31 + 6*T^30 + 12*T^29 + 11*T^28 + 10*T^27 + 3*T^26 + 12*T^25 + 12*T^24 + 9*T^23 + 11*T^21 + 4*T^19 + 4*T^18 + 8*T^17 + 10*T^16 + 5*T^15 + 7*T^14 + 4*T^13 + T^12 + 9*T^11 + 4*T^10 + 7*T^8 + 8*T^7 + 11*T^6 + 4*T^5 + 2*T^4 + 10*T^3 + 6, 10*T^33 + 11*T^32 + 8*T^31 + 12*T^30 + 9*T^29 + T^28 + 12*T^27 + 3*T^26 + T^25 + 6*T^24 + 7*T^23 + 7*T^22 + 2*T^21 + 4*T^20 + 10*T^19 + 4*T^18 + 3*T^17 + 7*T^16 + 3*T^15 + 7*T^14 + 2*T^12 + 3*T^11 + 11*T^10 + 5*T^9 + 11*T^7 + 2*T^6 + 9*T^5 + 10*T^4 + 8*T^3 + 2*T^2 + 3*T + 8, 7*T^33 + 9*T^32 + 8*T^31 + 4*T^30 + T^29 + 11*T^28 + 3*T^27 + 12*T^26 + 11*T^25 + 7*T^24 + 2*T^23 + 11*T^22 + 9*T^21 + 5*T^20 + 5*T^19 + 7*T^18 + 4*T^17 + 10*T^16 + 9*T^15 + 8*T^14 + 6*T^12 + 6*T^11 + 6*T^10 + 5*T^9 + 7*T^8 + 3*T^7 + 5*T^5 + 5*T^4 + 7*T^3 + 8*T^2 + T, 8*T^33 + 11*T^32 + 10*T^30 + 5*T^29 + T^28 + T^27 + T^26 + 2*T^25 + 6*T^23 + 12*T^22 + 11*T^21 + 2*T^20 + 8*T^19 + 10*T^18 + 7*T^17 + 7*T^16 + 2*T^15 + 5*T^14 + 12*T^13 + 11*T^12 + 7*T^11 + 2*T^10 + 7*T^9 + T^8 + 12*T^7 + 10*T^6 + 4*T^5 + 5*T^4 + 6*T^3 + 10*T^2 + 7*T + 2, 9*T^33 + 9*T^32 + 10*T^31 + 5*T^30 + 11*T^29 + 8*T^28 + 5*T^27 + 4*T^26 + 5*T^25 + 7*T^24 + 3*T^23 + T^22 + 6*T^21 + 8*T^20 + T^19 + 5*T^17 + 10*T^16 + 11*T^15 + T^14 + 9*T^13 + 12*T^12 + 6*T^11 + 7*T^10 + 7*T^9 + T^8 + 10*T^7 + 7*T^6 + 4*T^5 + 4*T^4 + T^3 + 8*T + 4 ] ####################################################### ####################################################### #### #### 8. BEISPIEL ####################################################### ####################################################### k := GF(13); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); kx := PolynomialAlgebra(k); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^5+2*y^3*T+3*y*T+T+1; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; h1 := t+Coerce(o,[3*T^2+2,T,1,1,2]); h2 :=t-Coerce(o,[4*T^4,T^2+1,T^5,1,4]); h3 :=t^2+Coerce(o,[T+1,1,0,0,0]); h4 :=t+Coerce(o,[3*T^11+T^10+3*T^5+2*T^7+T+1,T,1,T^11,T]); h5 :=t^2+Coerce(o,[T^2+1,T^2+1+T,1,1,0]); h6 :=t^2+Coerce(o,[T^11+3*T^3+1,2*T^2+1+T,T,3+T^2,0]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F);[9,33] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); *********************************** [ [ ot.1 + [ 3*T^2 + 2, T, 1, 1, 2 ] ], 4, 1500, 1 ] Time: 712.953948 s Das Polynom g hat folgende Gestalt: *********************************** ot.1^9 + [ 3*T^11 + T^10 + 2*T^7 + 3*T^5 + 9*T^4 + 3*T^2 + T + 3, 12*T^2 + 2*T + 12, 12*T^5 + 2, T^11, T + 11 ]*ot.1^8 + [ T^17 + T^16 + T^15 + 9*T^14 + 5*T^13 + 11*T^12 + 11*T^11 + 2*T^10 + 7*T^9 + 2*T^6 + 3*T^5 + 2*T^4 + 8*T^3 + T + 8, 3*T^17 + 11*T^13 + 12*T^12 + 11*T^11 + 12*T^10 + 11*T^9 + 2*T^8 + 9*T^7 + 9*T^6 + 4*T^5 + T^3 + T^2 + 3*T + 7, 10*T^16 + 12*T^15 + 5*T^11 + 11*T^10 + 2*T^7 + 5*T^6 + 5*T^4 + 12*T^3 + 5*T^2 + T + 10, 2*T^17 + 9*T^15 + 8*T^13 + 4*T^12 + 2*T^11 + 11*T^4 + 2*T^3 + T^2 + 10*T + 8, 12*T^13 + T^12 + 6*T^11 + 11*T^10 + 11*T^7 + 4*T^6 + T^5 + 5*T^4 + 8*T^3 + 11*T^2 + 11*T + 3 ]*ot.1^7 + [ 3*T^22 + T^21 + 11*T^19 + 5*T^18 + 10*T^17 + 11*T^16 + 5*T^15 + 6*T^14 + 7*T^13 + 5*T^12 + 6*T^10 + 7*T^9 + T^8 + 12*T^7 + 4*T^6 + 12*T^5 + 5*T^3 + 9*T^2 + 8*T + 5, 7*T^19 + T^18 + 11*T^17 + 2*T^16 + 8*T^15 + 8*T^14 + 6*T^13 + 11*T^12 + 4*T^11 + 12*T^10 + 2*T^9 + 10*T^8 + 5*T^7 + 9*T^6 + 6*T^5 + T^4 + 5*T^3 + 6*T^2 + 9*T + 7, 10*T^18 + 12*T^17 + 7*T^16 + 8*T^15 + 11*T^14 + 4*T^13 + 3*T^12 + 11*T^11 + 12*T^10 + 3*T^9 + 8*T^8 + 4*T^7 + 7*T^6 + 10*T^5 + 5*T^4 + 6*T^3 + 10*T^2 + T + 10, T^22 + 9*T^19 + 11*T^18 + 6*T^17 + 8*T^16 + 10*T^15 + 12*T^14 + T^13 + 8*T^12 + 8*T^11 + 4*T^10 + 6*T^9 + 11*T^8 + 2*T^7 + 5*T^6 + 8*T^5 + 7*T^4 + 12*T^3 + 6*T^2 + 4*T + 12, 11*T^18 + 4*T^17 + 7*T^16 + 3*T^15 + 10*T^14 + T^13 + 9*T^12 + 6*T^11 + T^10 + 12*T^9 + 12*T^8 + 12*T^6 + 8*T^5 + 8*T^4 + 7*T^3 + 3*T^2 + 7*T + 2 ]*ot.1^6 + [ T^28 + T^27 + T^26 + 9*T^25 + 5*T^24 + 11*T^23 + 10*T^22 + 2*T^21 + 10*T^20 + 5*T^19 + 5*T^18 + 8*T^17 + 6*T^16 + 8*T^15 + 4*T^14 + 8*T^13 + 11*T^12 + 4*T^11 + 9*T^10 + 12*T^9 + 10*T^7 + 11*T^6 + 2*T^5 + T^4 + 11*T^3 + 10*T^2 + 3*T + 5, 3*T^28 + 11*T^24 + 12*T^23 + 11*T^22 + 12*T^21 + 5*T^20 + 9*T^19 + 10*T^18 + 12*T^17 + 12*T^16 + 8*T^15 + 11*T^13 + T^12 + 4*T^10 + 12*T^8 + 10*T^7 + 8*T^6 + 3*T^5 + 3*T^4 + 3*T^3 + 6*T^2 + 12*T + 11, 10*T^27 + 12*T^26 + 5*T^22 + 11*T^21 + 7*T^20 + 4*T^18 + 12*T^17 + 4*T^16 + 10*T^15 + 7*T^14 + 11*T^13 + 4*T^12 + 11*T^11 + 5*T^10 + T^9 + 3*T^8 + 10*T^7 + 7*T^5 + 8*T^4 + T^3 + 7*T^2 + T + 11, 2*T^28 + 9*T^26 + 8*T^24 + 4*T^23 + 2*T^22 + 6*T^20 + 5*T^19 + 9*T^18 + 3*T^17 + 12*T^16 + 12*T^15 + 12*T^14 + 6*T^13 + 12*T^12 + 6*T^11 + 3*T^10 + 7*T^9 + 7*T^8 + 12*T^7 + 12*T^6 + 7*T^5 + 7*T^4 + 3*T^3 + 2*T^2 + 2*T + 8, 12*T^24 + T^23 + 6*T^22 + 11*T^21 + 9*T^20 + 9*T^19 + 7*T^18 + 7*T^17 + 2*T^16 + 10*T^15 + 11*T^14 + 4*T^13 + 10*T^12 + 9*T^11 + 10*T^10 + 2*T^9 + 7*T^8 + 3*T^7 + 9*T^6 + 11*T^5 + 7*T^4 + 8*T^3 + 5*T^2 + 9*T + 2 ]*ot.1^5 + [ 11*T^30 + 3*T^29 + 10*T^28 + 8*T^27 + 9*T^26 + 10*T^25 + 2*T^24 + 5*T^23 + 10*T^22 + 10*T^21 + 4*T^20 + 8*T^19 + 3*T^16 + 3*T^15 + 8*T^14 + 5*T^12 + 7*T^11 + 7*T^10 + 7*T^9 + 12*T^8 + 12*T^7 + T^6 + 10*T^5 + 5*T^4 + 7*T^3 + 5*T^2 + 7*T + 5, 7*T^30 + T^29 + 11*T^28 + 2*T^27 + 8*T^26 + 9*T^25 + 5*T^24 + 7*T^23 + 4*T^22 + 6*T^21 + 10*T^20 + 12*T^19 + 7*T^18 + 10*T^17 + 5*T^16 + 8*T^15 + 6*T^14 + 2*T^13 + 8*T^12 + 4*T^11 + 12*T^10 + 12*T^9 + 10*T^8 + 5*T^6 + 6*T^5 + 3*T^4 + 4*T^3 + 12*T + 4, 10*T^29 + 12*T^28 + 8*T^27 + 8*T^26 + T^25 + 4*T^24 + 9*T^23 + 7*T^22 + 6*T^21 + 7*T^20 + 12*T^19 + 8*T^18 + 4*T^17 + T^16 + 7*T^15 + 6*T^14 + 11*T^13 + 6*T^12 + 4*T^11 + 9*T^10 + 8*T^9 + 6*T^8 + 10*T^7 + T^6 + 2*T^5 + 3*T^3 + 8*T^2 + 10*T + 1, 9*T^30 + 11*T^29 + 6*T^28 + 8*T^27 + 10*T^26 + 9*T^25 + 7*T^23 + 4*T^22 + 11*T^21 + 12*T^20 + 7*T^19 + 10*T^18 + 5*T^17 + 8*T^16 + 5*T^14 + 4*T^13 + 2*T^12 + 10*T^11 + 8*T^10 + 10*T^9 + 8*T^8 + 8*T^7 + T^6 + 5*T^4 + 12*T^3 + 9*T^2 + 11, 11*T^29 + 4*T^28 + 7*T^27 + 3*T^26 + 12*T^25 + 12*T^24 + 2*T^22 + 7*T^21 + 7*T^20 + 4*T^19 + 8*T^18 + T^17 + 9*T^16 + 10*T^15 + 7*T^14 + 7*T^13 + 4*T^12 + 11*T^11 + 3*T^10 + 9*T^9 + 4*T^8 + 6*T^7 + 2*T^6 + 8*T^5 + 5*T^4 + 10*T^3 + 12*T^2 + 2*T + 5 ]*ot.1^4 + [ T^30 + 5*T^28 + 8*T^27 + 7*T^26 + 8*T^25 + 9*T^24 + 6*T^23 + 10*T^22 + 6*T^21 + 12*T^20 + 7*T^19 + 2*T^18 + 2*T^17 + 4*T^16 + 6*T^15 + 12*T^14 + 4*T^13 + 9*T^12 + 8*T^11 + 8*T^10 + 10*T^9 + T^7 + 4*T^6 + 5*T^5 + 7*T^4 + 9*T^3 + 11*T^2 + T + 12, 4*T^30 + 10*T^29 + 4*T^28 + 8*T^27 + 12*T^26 + 4*T^25 + 5*T^24 + 4*T^23 + 11*T^22 + 5*T^21 + 3*T^20 + 6*T^19 + 5*T^18 + 9*T^17 + 11*T^16 + 4*T^15 + 2*T^13 + 10*T^12 + 3*T^11 + 10*T^10 + 2*T^9 + 4*T^8 + 4*T^7 + 12*T^6 + 7*T^5 + 10*T^4 + 7*T^2 + 2, 3*T^30 + 7*T^29 + 3*T^28 + 6*T^27 + 10*T^26 + 12*T^25 + T^24 + 4*T^23 + T^22 + 10*T^21 + 3*T^20 + 3*T^19 + 6*T^18 + 10*T^17 + 8*T^16 + 4*T^15 + 12*T^14 + 9*T^13 + T^12 + 3*T^11 + 10*T^8 + 12*T^7 + 9*T^6 + T^5 + 2*T^4 + 5*T^3 + 9*T^2 + 7*T + 2, 2*T^30 + 8*T^29 + 6*T^28 + 2*T^27 + 8*T^26 + 2*T^25 + 5*T^24 + 10*T^23 + 3*T^22 + 8*T^21 + 11*T^20 + T^19 + 7*T^18 + 11*T^17 + 11*T^16 + 11*T^15 + 5*T^14 + 4*T^13 + 11*T^12 + 12*T^11 + 2*T^10 + T^9 + 3*T^8 + 8*T^7 + T^6 + 6*T^5 + 12*T^4 + 3*T^3 + 6*T + 2, 2*T^30 + 11*T^29 + 3*T^28 + T^27 + 11*T^26 + 11*T^25 + 2*T^24 + 12*T^23 + 10*T^22 + T^21 + 5*T^20 + 2*T^19 + 6*T^18 + 4*T^17 + 4*T^16 + 9*T^15 + T^14 + 10*T^13 + 8*T^12 + 12*T^11 + 2*T^10 + 12*T^9 + 9*T^8 + T^6 + 8*T^5 + 3*T^4 + 3*T^3 + 11*T^2 + 2*T + 3 ]*ot.1^3 + [ T^31 + 8*T^29 + 2*T^28 + 12*T^27 + 6*T^26 + T^25 + 2*T^24 + 12*T^23 + 10*T^22 + 7*T^21 + 8*T^19 + 8*T^18 + 6*T^17 + 3*T^16 + 7*T^15 + 9*T^14 + 9*T^13 + 6*T^12 + 4*T^11 + 5*T^10 + 11*T^9 + 2*T^8 + 10*T^7 + 4*T^6 + 9*T^5 + 8*T^4 + 8*T^3 + 4*T^2 + 4*T + 1, 11*T^32 + 7*T^31 + 9*T^30 + 6*T^29 + 5*T^28 + 11*T^27 + 10*T^25 + 8*T^24 + 3*T^23 + 2*T^21 + 6*T^20 + 6*T^19 + 12*T^18 + T^17 + 11*T^16 + 4*T^15 + T^14 + T^13 + 9*T^12 + 6*T^11 + 3*T^10 + 11*T^9 + 6*T^7 + 5*T^6 + 9*T^5 + 7*T^4 + 9*T^2 + 6*T + 7, 7*T^32 + 4*T^31 + 8*T^30 + 5*T^27 + 5*T^26 + 2*T^25 + 6*T^24 + 9*T^23 + 8*T^22 + 2*T^19 + 10*T^18 + 9*T^17 + 8*T^15 + 6*T^14 + 6*T^13 + 3*T^11 + 11*T^10 + 8*T^9 + 2*T^8 + 2*T^7 + 9*T^6 + 4*T^5 + 6*T^4 + 3*T^3 + 8*T^2 + 4*T, 2*T^30 + 11*T^29 + 5*T^28 + 2*T^27 + 8*T^26 + 6*T^25 + T^24 + 7*T^23 + 12*T^22 + 9*T^21 + 2*T^20 + 3*T^19 + T^18 + T^17 + 11*T^16 + 5*T^14 + 9*T^12 + T^11 + 10*T^10 + 7*T^9 + 6*T^8 + 12*T^7 + 5*T^6 + 8*T^5 + 10*T^4 + 4*T^3 + 5*T^2 + T + 6, 9*T^32 + 8*T^29 + 2*T^28 + 9*T^27 + 5*T^26 + 4*T^24 + 5*T^23 + 12*T^22 + 7*T^21 + 2*T^20 + 11*T^19 + 2*T^18 + T^17 + 3*T^16 + 4*T^15 + 7*T^14 + 3*T^13 + 8*T^12 + 6*T^11 + 8*T^10 + 7*T^8 + 11*T^7 + 2*T^6 + 5*T^5 + 5*T^4 + 11*T^3 + 2*T^2 + 12 ]*ot.1^2 + [ 12*T^31 + 3*T^29 + 7*T^28 + 2*T^26 + 6*T^25 + 3*T^24 + 2*T^23 + 5*T^22 + 2*T^21 + 2*T^20 + 11*T^19 + 7*T^18 + 11*T^16 + 10*T^15 + T^14 + 11*T^13 + 8*T^12 + 12*T^11 + 8*T^10 + 4*T^9 + T^8 + 11*T^7 + T^6 + 12*T^5 + 5*T^4 + 8*T^3 + 11*T^2 + 6, 11*T^31 + 5*T^30 + 3*T^29 + 7*T^28 + 5*T^27 + 5*T^26 + 11*T^25 + 10*T^24 + 5*T^23 + 5*T^22 + 12*T^21 + 6*T^20 + 8*T^19 + 7*T^18 + 9*T^17 + T^16 + 10*T^15 + 2*T^14 + 5*T^13 + 2*T^12 + 11*T^11 + 2*T^10 + 11*T^9 + 9*T^8 + 12*T^7 + T^6 + 10*T^5 + 10*T^4 + 4*T^3 + 12*T^2 + 12*T + 5, 3*T^31 + T^30 + 4*T^29 + 2*T^27 + 8*T^26 + 9*T^25 + 9*T^24 + T^23 + 4*T^22 + 10*T^21 + 5*T^19 + 5*T^18 + 2*T^17 + T^16 + 9*T^14 + 5*T^12 + 10*T^11 + 9*T^10 + 11*T^9 + 8*T^8 + 11*T^7 + T^5 + 8*T^4 + 2*T^3 + T + 12, 11*T^31 + 2*T^30 + 2*T^29 + T^28 + 2*T^27 + 5*T^26 + 11*T^25 + 12*T^24 + 12*T^21 + 4*T^20 + 11*T^19 + 5*T^18 + 9*T^17 + 7*T^16 + 7*T^15 + 9*T^14 + 6*T^12 + 6*T^11 + T^10 + 4*T^9 + 9*T^8 + 2*T^7 + 5*T^5 + 6*T^4 + 6*T^3 + 12*T^2 + 8*T, 2*T^31 + 2*T^30 + 5*T^29 + 6*T^28 + 12*T^27 + T^26 + 11*T^25 + 3*T^24 + 8*T^23 + 3*T^22 + 10*T^21 + 12*T^20 + 10*T^19 + T^18 + 8*T^16 + T^15 + 2*T^14 + 8*T^13 + 7*T^12 + 11*T^11 + 2*T^10 + 9*T^9 + 10*T^8 + 7*T^7 + 7*T^6 + 5*T^5 + 6*T^4 + 7*T^2 + 3*T ]*ot.1 + [ 4*T^33 + 7*T^32 + 7*T^31 + 6*T^30 + 12*T^29 + 11*T^28 + 10*T^27 + 3*T^26 + 12*T^25 + 12*T^24 + 9*T^23+ 11*T^21 + 4*T^19 + 4*T^18 + 8*T^17 + 10*T^16 + 5*T^15 + 7*T^14 + 4*T^13 + T^12 + 9*T^11 + 4*T^10 + 7*T^8 + 8*T^7 + 11*T^6 + 4*T^5 + 2*T^4 + 10*T^3 + 6, 10*T^33 + 11*T^32 + 8*T^31 + 12*T^30 + 9*T^29 + T^28 + 12*T^27 + 3*T^26 + T^25 + 6*T^24 + 7*T^23 + 7*T^22 + 2*T^21 + 4*T^20 + 10*T^19 + 4*T^18 + 3*T^17 + 7*T^16 + 3*T^15 + 7*T^14 + 2*T^12 + 3*T^11 + 11*T^10 + 5*T^9 + 11*T^7 + 2*T^6 + 9*T^5 + 10*T^4 + 8*T^3 + 2*T^2 + 3*T + 8, 7*T^33 + 9*T^32 + 8*T^31 + 4*T^30 + T^29 + 11*T^28 + 3*T^27 + 12*T^26 + 11*T^25 + 7*T^24 + 2*T^23 + 11*T^22 + 9*T^21 + 5*T^20 + 5*T^19 + 7*T^18 + 4*T^17 + 10*T^16 + 9*T^15 + 8*T^14 + 6*T^12 + 6*T^11 + 6*T^10 + 5*T^9 + 7*T^8 + 3*T^7 + 5*T^5 + 5*T^4 + 7*T^3 + 8*T^2 + T, 8*T^33 + 11*T^32 + 10*T^30 + 5*T^29 + T^28 + T^27 + T^26 + 2*T^25 + 6*T^23 + 12*T^22 + 11*T^21 + 2*T^20 + 8*T^19 + 10*T^18 + 7*T^17 + 7*T^16 + 2*T^15 + 5*T^14 + 12*T^13 + 11*T^12 + 7*T^11 + 2*T^10 + 7*T^9 + T^8 + 12*T^7 + 10*T^6 + 4*T^5 + 5*T^4 + 6*T^3 + 10*T^2 + 7*T + 2, 9*T^33 + 9*T^32 + 10*T^31 + 5*T^30 + 11*T^29 + 8*T^28 + 5*T^27 + 4*T^26 + 5*T^25 + 7*T^24 + 3*T^23 + T^22 + 6*T^21 + 8*T^20 + T^19 + 5*T^17 + 10*T^16 + 11*T^15 + T^14 + 9*T^13 + 12*T^12 + 6*T^11 + 7*T^10 + 7*T^9 + T^8 + 10*T^7 + 7*T^6 + 4*T^5 + 4*T^4 + T^3 + 8*T + 4 ] ################################################### ################################################### #### #### 9. BEISPIEL ################################################### ################################################### k := GF(7,4); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^3+2*y^2*T^2+y*T^11+T^2+T+21;; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T^2,T+1,2*T+2]); h2 :=t-Coerce(o,[T+1,T^3+1,T+2]); h3 :=t^2+Coerce(o,[1,T,2+T^2]); h4 := t+Coerce(o,[T+1,2,T^6]); h5 := t-Coerce(o,[T^2+T,T^3+1,T^2+6]); h6:= t^3+Coerce(o,[1+T,T^2,2+T]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F);[9,70] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ t + [ T + 1, 2, T^6 ] ], 6, 408, 1 ]---->>> Time: 1325.756981 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ 6*T, 5*T^3 + T + 1, T^6 + 6*T^2 + T + 1 ]*t^8 + [ 5*T^12 + 2*T^11 + 6*T^10 + T^9 + 5*T^7 + 3*T^6 + 2*T^4 + 4*T^3 + 4*T^2, 5*T^21 + 4*T^20 + 2*T^19 + 6*T^18 + T^17 + 4*T^16 + 6*T^15 + T^14 + 2*T^13 + 3*T^12 + 2*T^11 + T^10 + 5*T^8 + 6*T^7 + 3*T^4 + 5*T^3 + 6*T^2 + 4, T^19 + 6*T^18 + 6*T^17 + T^14 + 2*T^13 + T^12 + 5*T^11 + 4*T^10 + 5*T^9 + 2*T^8 + 6*T^7 + 5*T^6 + 2*T^5 + T^4 + 2*T^2 + 4*T + 4 ]*t^7 + [ 5*T^24 + 3*T^23 + 3*T^21 + 3*T^20 + 6*T^19 + 5*T^18 + 5*T^16 + 6*T^15 + 6*T^14 + 2*T^13 + 4*T^12 + 2*T^11 + T^10 + 6*T^9 + T^8 + T^7 + 6*T^6 + 3*T^5 + T^4 + T^3 + 5*T^2 + 2*T + 1, 5*T^33 + 5*T^32 + 2*T^31 + T^30 + 2*T^29 + 4*T^28 + T^27 + 6*T^26 + 6*T^25 + 6*T^23 + 2*T^22 + 5*T^21 + T^20 + 5*T^19 + 2*T^18 + 4*T^17 + 6*T^14 + T^13 + 6*T^12 + 2*T^11 + 2*T^10 + 4*T^8 + 3*T^7 + 2*T^6 + 5*T^5 + 3*T^4 + T^3 + 2*T^2 + 5*T + 1, 6*T^31 + 5*T^30 + 2*T^29 + 3*T^28 + 2*T^26 + 6*T^25 + 4*T^24 + 5*T^23 + 5*T^22 + 5*T^21 + 2*T^20 + 5*T^19 + 5*T^18 + 6*T^17 + 6*T^16 + T^14 + T^13 + 4*T^12 + 4*T^11 + 2*T^9 + 3*T^8 + 3*T^6 + 4*T^5 + 5*T^4 + T^3 + 3*T^2 + T ]*t^6 + [ 3*T^36 + 4*T^34 + 6*T^33 + 5*T^32 + 5*T^31 + 6*T^30 + 2*T^29 + 5*T^27 + 3*T^26 + 5*T^25 + 2*T^24 + T^23 + 5*T^22 + 4*T^21 + T^20 + 5*T^18 + 5*T^16 + T^13 + 5*T^12 + T^11 + 2*T^10 + 5*T^9 + 6*T^8 + 5*T^7 + 6*T^5 + 3*T^4 + 5*T^2 + 6*T + 6, 3*T^45 + 4*T^44 + 6*T^42 + 6*T^41 + 6*T^40 + 2*T^38 + 5*T^37 + 3*T^35 + 6*T^33 + T^31 + 3*T^30 + 5*T^29 + 6*T^28 + T^27 + 5*T^26 + 6*T^25 + 4*T^24 + 4*T^23 + 3*T^22 + 3*T^21 + 6*T^20 + 2*T^19 + 5*T^18 + 6*T^17 + 4*T^16 + T^15 + 5*T^14 + 5*T^13 + 4*T^12+ 2*T^11 + T^10 + 3*T^9 + 5*T^8 + 2*T^7 + 5*T^5 + 4*T^4 + 6*T^3 + 5*T^2 + 4*T + 5, 5*T^43 + T^42 + 5*T^41 + 6*T^40 + 4*T^39 + T^36 + 3*T^35 + 6*T^33 + 4*T^32 + 5*T^31 + 2*T^30 + T^29 + 6*T^28 + 6*T^27 + 3*T^26 + 3*T^25 + T^24 + 4*T^22 + 6*T^21 + 3*T^19 + 5*T^18 + 6*T^17 + 4*T^15 + 2*T^13 + 6*T^12 + T^11 + 2*T^10 + 3*T^9 + 5*T^8 + T^7 + 2*T^6 + 2*T^5 + 3*T^4 + 3*T^3 + 4*T^2 + 2 ]*t^5 + [ 5*T^36 + 4*T^34 + 6*T^33 + 5*T^32 + 5*T^31 + 6*T^30 + 5*T^29 + 4*T^28 + 2*T^27 + 4*T^26 + T^25 + T^24 + 5*T^23 + 3*T^22 + 6*T^21 + 5*T^20 + 4*T^19 + 4*T^17+ 3*T^16 + 4*T^15 + 5*T^14 + 3*T^13 + 6*T^12 + T^11 + 6*T^10 + 3*T^9 + 2*T^8 + 3*T^7 + T^5 + 5*T^4 + T^3 + 3*T, 5*T^45 + 2*T^44 + 2*T^43 + 4*T^42 + T^41 + 4*T^40 + 2*T^39 + 3*T^38 + T^37 + T^36 + 4*T^35 + T^34 + 5*T^33 + 3*T^32 + 5*T^31 + 4*T^28 + 6*T^27 + 6*T^26 + 6*T^24 + 3*T^23 + 4*T^22 + 3*T^21 + 2*T^20 + 2*T^19 + 3*T^18 + 4*T^17 + 5*T^16 + 6*T^15 + 5*T^14 + T^12 + 2*T^11 + 3*T^9 + 3*T^8 + 6*T^7 + 3*T^6 + 5*T^5 + T^4 + T^3 + 2*T^2 + T + 4, T^44 + 2*T^43 + T^41 + 3*T^40 + 6*T^39 + T^38 + 6*T^37 + 5*T^35 + T^34 + T^33 + 2*T^32 + 5*T^31 + 4*T^30 + 6*T^29 + T^27 + 6*T^26 + 2*T^25 + 5*T^24 + 5*T^23 + 4*T^22 + 3*T^21 + 5*T^20 + 6*T^19 + T^18 + 6*T^17 + 3*T^15 + 4*T^14 + 2*T^13 + 3*T^12 + 5*T^11+ 4*T^10 + 4*T^8 + 2*T^7 + T^6 + 5*T^5 + T^4 + 3*T^3 + 2*T + 1 ]*t^4 + [ 5*T^48 + T^47 + 4*T^45 + T^44 + 6*T^43 + 3*T^42 + 6*T^41 + T^40 + 3*T^39 + 4*T^37 + 3*T^35 + 2*T^34 + 3*T^33 + 3*T^32 + 2*T^31 + 5*T^30 + 3*T^29 + 2*T^28 + 6*T^27 + 5*T^26 + 3*T^25 + 6*T^23 + 3*T^22 + 4*T^21 + 5*T^20 + 5*T^19 + T^18 + 4*T^17 + 4*T^16 + 3*T^15 + T^14 + 6*T^13 + T^11 + 6*T^10 + 4*T^9 + 5*T^7 + T^6 +5*T^5 + 6*T^4 + T^3 + T^2 + 2*T, 5*T^57 + 3*T^56 + 4*T^55 + T^53 + 5*T^52 + 5*T^51 + T^50 + 3*T^48 + 6*T^47 + T^46 + 5*T^45 + T^44 + 5*T^43 + 6*T^42 + T^40 + 5*T^39 + 3*T^38 + 4*T^37 + 2*T^36 + 4*T^35 + 2*T^32 + 5*T^31 + 5*T^30 + T^29 + T^28 + 2*T^26 + 6*T^25 + 3*T^23 + 4*T^22 + 5*T^21 + 6*T^20 + 6*T^19 + 6*T^18 + 4*T^17 + 3*T^15 + 3*T^14 + 4*T^13 + 3*T^12 + 4*T^11 + 6*T^10 + T^8 + 3*T^7 + 4*T^6 + 2*T^5 + 2*T^4 + 2*T^3 + 2*T^2 + 4*T + 1, 2*T^56 + 6*T^55 + 6*T^54 + 6*T^53 + 2*T^51 + 6*T^50 + 6*T^49 + 4*T^47 + 6*T^46 + 6*T^45 + 2*T^43 + 5*T^41 + T^40 + 2*T^38 + 5*T^36 + 4*T^35 + 3*T^34 + 5*T^33 + 2*T^32 + 3*T^31 + 4*T^30 + T^29 + T^27 + 2*T^25 + 2*T^23 + 6*T^20 + 5*T^18 + 3*T^17 + 2*T^16 + T^15 + 3*T^14 + 3*T^13 + 4*T^12 + 5*T^11 + 5*T^10 + 3*T^9 + 2*T^8 + 5*T^7 + 2*T^6 + 6*T^5 + 6*T^4 + 4*T^3 + 5*T^2 + 5*T + 6 ]*t^3 + [ 5*T^46 + 5*T^45 + T^44 + 2*T^43 + 3*T^42 + 2*T^41 + 6*T^39 + T^38 + 4*T^37 + 4*T^36 + 6*T^35 + 3*T^34 + 4*T^32 + 3*T^31 + 4*T^30 + T^29 + 3*T^28 + T^27 + 5*T^26 + 5*T^25 + 5*T^22 + 4*T^21 + 3*T^19 + 5*T^18 + 3*T^17 + 4*T^16 + 6*T^15 + 2*T^14 + 6*T^13 + 3*T^12 + 5*T^10 + 3*T^9 + T^7 + 5*T^6 + 6*T^5 + T^4 + 2*T^3 + 6, 5*T^55 + T^53 + T^52 + 2*T^51 + 6*T^48 + 2*T^47 + 4*T^46 + 4*T^43 + 4*T^42 + 6*T^40 + 3*T^39 + T^38 + 2*T^37 + T^36 + 6*T^35 + 3*T^34 + 4*T^33 + T^32 + 6*T^31 + 3*T^30 + 3*T^29 + 6*T^28 + 3*T^27 + T^26 + 6*T^24 + 3*T^23 + 2*T^22 + 2*T^21 + 5*T^20 + T^19 + 5*T^17 + 3*T^16 + 6*T^15 + 3*T^14 + 5*T^9 + T^8 + 3*T^7 + 4*T^6 + 4*T^5 + 5*T^4 + 5*T^3 + 2*T^2 + 6*T + 4, 2*T^55 + 3*T^54 + 5*T^52 + 5*T^51 + 6*T^50 + 6*T^48 + 5*T^47 + T^46 + T^45 + 4*T^44 + T^43 + 4*T^42 + 6*T^41 + 4*T^40 + 3*T^39 + 3*T^38 + 3*T^37 + 4*T^35 + 2*T^34 + 6*T^33 + 4*T^32 + 6*T^31 + T^30 + T^28 + T^27 + 3*T^26 + 6*T^24 + 5*T^23 + 4*T^22 + 5*T^21 + 4*T^20 + 4*T^19 + 4*T^18 + 2*T^17 + T^16 + 4*T^14 + 4*T^13 + 2*T^12 + T^11 + 6*T^10 + 2*T^9 + T^8 + 4*T^6 + 6*T^5 + 6*T^4 + 3*T^3 + 6*T^2 + 4*T + 6 ]*t^2 + [ 2*T^49 + 4*T^48 + 3*T^47 + 4*T^46 + 4*T^45 + 6*T^44 + 5*T^43 + 5*T^42 + 2*T^41 + 3*T^39 + 6*T^38 + 2*T^37 + 4*T^36 + T^35 + 3*T^34 + 2*T^33 + 3*T^32 + 5*T^30 + 5*T^29 + T^28 + T^26 + 4*T^23 + 2*T^22 + T^21 + T^19 + 4*T^18 + 2*T^17 + 5*T^16 + T^15 + 5*T^14 + 2*T^13 + 5*T^12 + 6*T^11 + 5*T^10 + 2*T^9 + 3*T^8 + 6*T^6 + 6*T^5 + 5*T^4 +5*T^3 + 2*T^2 + 4*T, 2*T^58 + 2*T^57 + T^56 + 3*T^55 + T^54 + 5*T^53 + 5*T^51 + 4*T^50 + 3*T^49 + 5*T^47 + 5*T^46 + 2*T^44 + 4*T^43 + 3*T^42 + 3*T^40 + 3*T^39 + 5*T^38 + 3*T^37 + 2*T^35 + 4*T^34 + T^33 + 5*T^31 + 2*T^30 + 6*T^29 + 4*T^28 + 2*T^27 + T^26 + 4*T^25 + 6*T^24 + 4*T^23 + 2*T^22 + 3*T^21 + T^20 + 2*T^19 + 2*T^18 + 6*T^17 + 2*T^16 + 4*T^15 + 4*T^14 + 4*T^13 + 3*T^12 + 2*T^11 + 3*T^10 + 5*T^9 + 5*T^8 + 6*T^6 + 2*T^5 + 6*T^3 + 4*T, 6*T^56 + 3*T^55 + 3*T^54 + 6*T^53 + 2*T^52 + 2*T^51 + T^50 + 3*T^49 + 6*T^48 + 2*T^47 + 4*T^46 + 2*T^45 + 6*T^44 + 5*T^42 + T^41 + 4*T^39 + 4*T^38 + 5*T^37 + 4*T^36 + T^35 + 5*T^33 + 2*T^32 + 2*T^31 + 6*T^29 + T^28 + 5*T^27 + 5*T^26 + 4*T^25 + 6*T^23 + 4*T^21 + 5*T^19 + 6*T^18 + 3*T^17 + 3*T^16 + 3*T^15 + 3*T^13 + 6*T^12 + 4*T^10 + 5*T^9 + 6*T^8 + 5*T^7 + 3*T^6 + 3*T^5 + 6*T^4 + 3*T^3 + 5*T^2 + 6*T + 4 ]*t + [ 4*T^61 + 3*T^60 + 2*T^59 + 2*T^58 + 2*T^57 + 6*T^56 + 5*T^54 + T^53 + 6*T^52 + 6*T^51 + 4*T^50 + 6*T^48 + 6*T^45 + 3*T^44 + 6*T^43 + 3*T^42 + 2*T^41 + 3*T^40 + 6*T^38 + 2*T^37 + 2*T^36 + 3*T^34 + 3*T^33 + 6*T^32 + 4*T^31 + 3*T^29 + T^28 + 4*T^27 + 3*T^24 + 4*T^23 + 6*T^22 + T^21 + 6*T^20 + 6*T^19 + 3*T^18 + 4*T^17 + 3*T^16 + 3*T^15 + 2*T^13 + T^12 + 5*T^11 + 4*T^10 + 2*T^9 + 4*T^8 + 5*T^6 + 4*T^5 + 2*T^4 + 5*T^3 + 2*T^2 + 3*T, 4*T^70 + 6*T^69 + 3*T^68 + 6*T^67 + 3*T^66 + 3*T^65 + 4*T^64 + T^63 + 6*T^61 + 2*T^59 + 5*T^58 + 3*T^57 + 3*T^56 + T^55 + 3*T^54 + 2*T^53 + 6*T^52 + 2*T^51 + T^50 + 2*T^49 + 2*T^47 + 6*T^46 + 3*T^44 + 5*T^43 + 6*T^42 + 2*T^41 + 3*T^40 + 5*T^39 + 3*T^38 + 2*T^37 + 3*T^36 + 3*T^35 + 5*T^33 + 3*T^32 + T^31 + 3*T^30 + 3*T^29 + 5*T^27 + 3*T^26 + 2*T^25 + 3*T^24 + 2*T^23 + 2*T^22 + T^21 + T^20 + 5*T^19 + T^18 + 2*T^17 + 3*T^16 + 4*T^15 + 6*T^14 + T^13 + 2*T^12 + 4*T^11 + 3*T^10 + 2*T^9 + 6*T^8 + 4*T^7 + 6*T^6 + 6*T^5 + T^4 + 6*T^3 + 6*T^2 + 6*T, 5*T^68 + 4*T^67 + 3*T^66 + 3*T^65 + 2*T^64 + 5*T^63 + 4*T^62 + 4*T^61 + T^58 + T^57 + 2*T^56 + 6*T^55 + 2*T^53 + T^52 + 2*T^49 + 4*T^47 + 3*T^46 + T^45 + 5*T^44 + 5*T^43 + T^42 + 3*T^41 + T^40 + 4*T^39 + 3*T^38 + 3*T^37 + 3*T^35 + 6*T^34 + 2*T^33 + T^30 + 2*T^29 + 2*T^28 + 6*T^27 + 3*T^26 + T^25 + 4*T^24 + 6*T^23 + 6*T^22 + 4*T^21 + T^20 + 6*T^19 + 3*T^18 + 3*T^17 + 3*T^16 + 3*T^15 + 3*T^14 + 3*T^13 + T^11 + 4*T^10 + T^8 + 2*T^7 + 6*T^6 + T^5 + 5*T^3 + 4*T^2 + 5*T + 1 ] #################################################### #################################################### #### #### 10. BEISPIEL #################################################### #################################################### k := GF(5,6); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^4+2*y^3*T^2+2*y^2*T^2+T^3+T^2+1; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T,T,1,0]); h2 :=t-Coerce(o,[1,T,T+1,0]); h3 := t+Coerce(o,[T+1,1,T,0]); h4 :=t^2+Coerce(o,[1,7*T,T+1,0]); h5 := t+Coerce(o,[T^2,T+4,T^2+T^3,2+2*T]); h6:=t^3+ Coerce(o,[T+1,T^2+4,T^4+T^3,3+3*T]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F);[ 9, 29 ] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [[ t + [ 4, 4*T, 4*T + 4, 0 ] ], 4, 225, 1 ]--->>> Time: 480.150735s [ [ t + [ T + 1, 1, T, 0 ] ], 4, 225, 1 ]--->>>Time: 487.382369 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ T^2 + 2*T, T, T^3 + T^2, 2*T + 2 ]*t^8 + [ T^5 + T^2 + 3*T + 4, 2*T^2 + 4, 4*T^4 + T^2 + 4*T + 4, 2*T^4 + 4*T^3 + 2*T^2 + T ]*t^7 + [ 3*T^12 + 4*T^11 + 2*T^10 + 3*T^9 + T^8 + 3*T^7 + 4*T^6 + 4*T^5 + T^4 + T^3 + T^2 + 2*T + 4, T^10 + 3*T^9 + 2*T^8 + 3*T^6 + 3*T^5 + 4*T^4 + 4*T^3 + T^2 + 4*T, T^11 + 2*T^10 + 2*T^9 + 3*T^7 + 2*T^6 + 3*T^5 + 4*T^4 + 3*T^3 + 2*T^2, T^11 + 2*T^10 + 4*T^9 + 2*T^8 + 2*T^7 + 2*T^6 + 2*T^5 + 3*T^4 + 4*T^3 ]*t^6 + [ 2*T^16 + T^14 + T^13 + 4*T^12 + T^11 + 3*T^10 + 4*T^8 + T^7 + 2*T^6 + T^5 + 3*T^4 + T^3 + T^2 + T + 4, 4*T^14 + 4*T^12 + 2*T^11 + T^10 + 2*T^9 + 3*T^8 + 3*T^7 + 3*T^6 + 2*T^5 + T^3, 4*T^15 + T^14 + T^13 + 3*T^10 + 2*T^9 + 2*T^8 + 3*T^7 + 2*T^6 + 4*T^5 + 2*T^4 + 2*T^2 + 4*T + 1, 4*T^15 + T^14 + 4*T^13 + 4*T^12 + T^11 + 2*T^10 + 3*T^9 + 2*T^8 + 4*T^7 + 3*T^6 + T^5 + 2*T^4 + 3*T^3 + 4*T^2 + 4 ]*t^5 + [ 2*T^17 + 3*T^16 + T^14 + T^13 + 3*T^11 + 2*T^10 + T^9 + T^8 + 2*T^7 + 2*T^6 + T^5 + 2*T^4 + 4*T^3 + T^2 + T, 4*T^15 + T^14 + 2*T^13 + 4*T^11 + 3*T^10 + T^9 + 4*T^8 + 3*T^7 + 2*T^6 + 4*T^5 + 4*T^4 + 2*T^3 + 2*T^2 + T + 1, 4*T^16 + 2*T^15 + 3*T^14 + 3*T^13 + 2*T^12 + 4*T^9 + 3*T^8 + T^7 + 3*T^5 + 2*T^4 + 4*T^3 + 3*T^2 + 3, 4*T^16 + 2*T^15 + T^14 + 4*T^13 + 3*T^9 + T^7 + T^5 + T^3 + T^2 + 2*T + 1 ]*t^4 + [ 3*T^21 + 4*T^19 + 4*T^18 + 2*T^16 + 4*T^15 + 4*T^13 + 3*T^12 + 2*T^11 + 4*T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + 2*T^3 + 2*T + 3, T^19 + T^17 + 3*T^16 + 2*T^15 + 4*T^14 + 3*T^13 + 2*T^12 + 4*T^10 + T^9 + 3*T^8 + 3*T^7 + 2*T^6 + 3*T^5 + 4*T^4 + 2*T^3 + 2*T^2 + 2*T + 1, T^20 + 4*T^19 + 4*T^18 + 3*T^16 + 4*T^14 + 3*T^11 + T^9 + 3*T^8 + 4*T^7 + 4*T^6 + 4*T^4 + 4*T^3 + 2*T^2 + T + 1, T^20 + 4*T^19 + T^18 + T^17 + 2*T^16 + T^15 + 4*T^14 + 2*T^12 + 3*T^11 + 3*T^10 + 3*T^9 + 3*T^8 + 2*T^7 + 3*T^6 + 2*T^4 + 2*T^3 + 3*T^2 + 3*T ]*t^3 + [ 3*T^24 + 3*T^22 + 3*T^21 + 4*T^20 + 4*T^18 + 4*T^17 + 3*T^14 + T^13 + T^12 + 4*T^11 + 4*T^10 + 4*T^9 + 3*T^8 + 4*T^7 + 3*T^6 + 2*T^2 + 4*T + 4, T^22 + 4*T^20 + T^19 + 4*T^18 + 4*T^17 + T^16 + 3*T^14 + 3*T^13 + 2*T^12 + 2*T^11 + 2*T^10 + 4*T^9 + T^8 + 2*T^7 + T^6 + 2*T^5 + 4*T^4 + T^3 + 3*T^2 + 4*T + 2, T^23 + 4*T^22 + 2*T^21 + T^19 + T^18 + 2*T^16 + 3*T^13 + T^12 + 4*T^8 + 3*T^6 + 2*T^5 + T^4 + 4*T^3 + 3*T^2 + 2*T + 1, T^23 + 4*T^22 + 4*T^21 + T^20 + T^19 + T^18 + 3*T^17 + 2*T^16 + 2*T^15 + T^14 + 4*T^13 + T^11 + T^10 + T^9 + 3*T^7 + T^6 + 3*T^4 + T^3 + 2*T^2 + T + 1 ]*t^2 + [ 3*T^25 + 2*T^24 + 4*T^23 + 4*T^22 + T^19 + 4*T^17 + 3*T^16 + T^15 + 2*T^14 + 2*T^13 + 2*T^12 + 4*T^11 + 3*T^10 + 3*T^9 + 2*T^7 + 3*T^6 + T^4 + 3*T^2 + T + 2, T^23 + 4*T^22 + T^21 + 2*T^19 + 2*T^18 + T^17 + 4*T^16 + 2*T^14 + 3*T^12 + 3*T^11 + 4*T^10 + T^9 + 4*T^8 + 3*T^7 + 3*T^6 + 3*T^5 + 3*T^3 + 4*T^2 + 3*T + 1, T^24 + 3*T^23 + 4*T^21 + 3*T^20 + 3*T^19 + 4*T^18 + T^17 + 3*T^16 + 3*T^15 + 2*T^13 + 3*T^12 + T^10 + 3*T^9 + 3*T^8 + T^6 + 4*T^4 + 4*T^3 + 2*T^2 + 3*T + 1, T^24 + 3*T^23 + 2*T^22 + 3*T^21 + T^20 + T^19 + 2*T^18 + 2*T^16 + 4*T^15 + 4*T^14 + 3*T^13 + 4*T^12 + 4*T^11 + 2*T^9 + 3*T^8 + T^7 + 4*T^6 + 2*T^4 + 3*T^3 + T^2 + T + 4 ]*t + [ 2*T^29 + 2*T^28 + 3*T^27 + 2*T^26 + 2*T^25 + T^24 + 3*T^23 + 4*T^22 + 4*T^21 + T^20 + T^19 + 3*T^18 + 4*T^17 + T^16 + 4*T^13 + 2*T^12 + 4*T^11 + 2*T^10 + 2*T^8 + T^7 + 2*T^6 + 2*T^4 + T^3 + T + 1, 4*T^27 + 4*T^26 + 3*T^25 + T^24 + 4*T^23 + 4*T^21 + 3*T^20 + 2*T^19 + 3*T^18 + 3*T^17 + 2*T^16 + T^15 + 2*T^14 + 3*T^13 + T^12 + 3*T^10 + 2*T^9 + 3*T^8 + T^7 + 4*T^6 + 2*T^5 + T^4 + 4*T^3 + 3*T^2 + 3, 4*T^28 + T^26 + 2*T^25 + 3*T^24 + T^23 + 2*T^22 + 2*T^21 + 3*T^20 + 2*T^19 + 4*T^18 + 2*T^16 + 3*T^15 + T^14 + T^13 + 4*T^12 + 3*T^8 + 2*T^7 + 4*T^6 + T^5 + 2*T^4 + T^3 + 3*T^2 + 4*T + 3, 4*T^28 + 4*T^26 + 4*T^25 + T^24 + 2*T^22 + 4*T^21 + T^20 + 4*T^19 + T^17 + 4*T^16 + T^15 + 3*T^14 + 4*T^12 + 4*T^11 + 2*T^9 + T^7 + 3*T^5 + 4*T^4 + T^3 + 4*T^2 + 4 ] ####################################################### ####################################################### #### #### 11. BEISPIEL ####################################################### ####################################################### k := GF(7,3); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^11+11*y^5*T^3+3*y^3*T^7+4*T*y^2+T^11+T^7+12*T^2+7*T+4; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[T^12+T,T+1,0,0,0,0,0,0,0,0,0]); h2 := t^3-Coerce(o,[2*T+2,1,2*T,0,0,0,0,0,0,0,0]); h3 := t+Coerce(o,[T^2+T,T+1,0,0,0,0,0,0,0,0,0]); h4 := t^3+Coerce(o,[4*T^15+T^3+8,2*T^2+3*T+7,T^3+1,3,0,0,0,0,0,0,0]); g := h1*h2*h3*h4;; Coefgr(g,f,F);[ 9, 30 ] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ************************************ [ [ t + [ T^2 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ], 3, 528, 1 ] Time: 837.512477 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ T^2 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]*t^8 + [ T^12 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]*t^7 + [ 4*T^15 + T^14 + T^13 + 2*T^3 + T^2 + 5*T + 6, T^13 + T^12 + T^3 + 5*T^2 + 5*T + 6, T^3 + T^2 + 2, 3, 0, 0, 0, 0, 0, 0, 0 ]*t^6 + [ 4*T^17 + 4*T^16 + T^5 + T^4 + 5*T^3 + 4*T^2 + 6*T, 4*T^16 + 4*T^15 + 3*T^4 + 6*T^3 + 3*T + 6, T^5 + T^4 + 4*T^2 + 3*T + 6, T^4 + T^3 + T^2 + 2*T + 1, 3*T + 3, 0, 0, 0, 0, 0, 0 ]*t^5 + [ 4*T^27 + 4*T^16 + T^15 + 5*T^13 + 6*T^12 + T^4 + 5*T^2 + 6*T, 4*T^16 + 4*T^15 + 2*T^14 + 3*T^13 + 6*T^12 + T^4 + 3*T^3 + T^2 + 3*T + 6, T^15 + 5*T^13 + T^12 + T^4 + 2*T^3 + 3*T^2 + 3*T + 6, 3*T^12 + T^4 + T^3 + 5*T^2 + 2*T + 1, 3*T + 3, 0, 0, 0, 0, 0, 0 ]*t^4 + [ 4*T^29 + 4*T^28 + 4*T^18 + 5*T^17 + 4*T^15 + 4*T^14 + 6*T^13 + T^6 + T^5 + 3*T^4 + 2*T^3 + 6*T^2 + 5*T + 5, 4*T^28 + 4*T^27 + 4*T^18 + 5*T^17 + 4*T^16 + 2*T^15 + 3*T^13 + 6*T^12 + T^6 + 5*T^5 + 5*T^4 + 4*T^3 + 3*T^2 + 6*T + 6, 5*T^17 + T^16 + 4*T^15 + 4*T^14 + 3*T^13 + 6*T^12 + T^6 + 4*T^5 + 5*T^4 + T^3 + 4*T^2 + T + 4, T^16 + T^15 + T^14 + 2*T^13 + T^12 + T^6 + 3*T^5 + 2*T^4 + 3*T^2 + 4*T + 6, 3*T^13 + 3*T^12 + T^5 + 2*T^3 + 6*T^2 + 4*T + 5, 3*T^2 + 3, 0, 0, 0, 0, 0 ]*t^3 + [ 6*T^18 + 5*T^17 + 6*T^16 + 5*T^6 + 3*T^5 + 5*T^4 + 5*T^3 + 3*T^2 + 5*T, 2*T^17 + T^16 + 6*T^15 + 2*T^4 + 3*T^3 + 5*T^2 + 2*T + 5, 6*T^18 + 6*T^17 + 3*T^16 + 3*T^15 + 3*T^6 + T^5 + 5*T^4 + 4*T^3 + 3*T^2 + 5*T + 6, 6*T^17 + 6*T^16 + 5*T^5 + 4*T^4 + 5*T^3 + 6*T^2 + 5*T + 5, 5*T^6 + 5*T^5 + 2*T^4 + T^3 + 4*T^2 + 5*T, 5*T^5 + 5*T^4 + T^3 + 6*T^2 + 2*T + 4, T^2 + T, 0, 0, 0, 0 ]*t^2 + [ 6*T^28 + 6*T^27 + 6*T^17 + 4*T^16 + 5*T^15 + 5*T^13 + 5*T^12 + 5*T^5 + 5*T^4 + 5*T^2 + 5*T, 3*T^27 + 6*T^17 + T^16 + T^15 + 4*T^14 + T^13 + 6*T^12 + 5*T^5 + 5*T^4 + 2*T^3 + 6*T^2 + 2*T + 5, 6*T^28 + 6*T^17 + 6*T^16 + T^15 + 5*T^14 + 5*T^12 + 3*T^5 + 4*T^3 + 5*T^2 + 5*T + 6, 6*T^17 + 6*T^16 + 2*T^15 + T^14 + T^13 + 3*T^5 + 3*T^4 + 4*T^3 + 6*T^2 + 5*T + 5, 5*T^16 + 5*T^13 + 4*T^12 + 5*T^5 + 2*T^4 + 3*T^3 + 5*T, T^13 + 5*T^5 + 5*T^4 + 6*T^2 + 2*T + 4, T^2 + T, 0, 0, 0, 0 ]*t + [ 6*T^30 + 5*T^29 + 6*T^28 + 6*T^19 + 3*T^18 + 2*T^17 + 5*T^16 + 5*T^15 + 3*T^14 + 5*T^13 + 5*T^7 + 3*T^6 + 5*T^5 + 5*T^4 + 3*T^3 + 5*T^2, 2*T^29 + T^28 + 6*T^27 + 6*T^19 + 6*T^18 + 5*T^17 + 3*T^15 + 5*T^14 + 2*T^13 + 5*T^12 + 5*T^7 + T^6 + 3*T^5 + 6*T^4 + 6*T^3 + 3*T^2 + 3*T, 6*T^30 + 6*T^29 + 3*T^28 + 3*T^27 + 6*T^19 + 4*T^18 + T^16 + 3*T^15 + 3*T^14 + 5*T^13 + 6*T^12 + 3*T^7 + T^6 + 2*T^4 + 4*T^3 + 5*T^2 + 6*T + 5, 6*T^29 + 6*T^28 + 6*T^19 + 4*T^18 + 6*T^17 + 3*T^16 + T^15 + 6*T^14 + 5*T^13 + 5*T^12 + 3*T^7 + 2*T^6 + 3*T^5 + 6*T^3 + 6*T^2 + 2*T + 6, 4*T^18 + 3*T^17 + T^16 + T^15 + 4*T^14 + 5*T^13 + 5*T^7 + 3*T^6 + 4*T^5 + 3*T^4 + T^3 + 2*T^2 + 3*T + 5, 5*T^17 + 5*T^16 + T^15 + 6*T^14 + 2*T^13 + 4*T^12 + 5*T^7 + T^6 + 5*T^5 + 4*T^4 + 4*T^3 + 4*T^2 + 2*T, T^14 + T^13 + 5*T^6 + 3*T^5 + 6*T^4 + T^3 + 2*T^2 + 6*T + 4, T^3 + 2*T^2 + T, 0,0, 0 ] ####################################################### ####################################################### #### #### 12. BEISPIEL ####################################################### ####################################################### k := GF(7,5); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^11+11*y^5*T^3+3*y^3*T^7+4*T*y^2+T^11+T^7+12*T^2+7*T+4; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[T^12+T,T+1,0,0,0,0,0,0,0,0,0]); h2 := t^3-Coerce(o,[2*T+2,1,2*T,0,0,0,0,0,0,0,0]); h3 := t+Coerce(o,[T^2+T,T+1,0,0,0,0,0,0,0,0,0]); h4 := t^3+Coerce(o,[4*T^15+T^3+8,2*T^2+3*T+7,T^3+1,3,0,0,0,0,0,0,0]); g := h1*h2*h3*h4;; Coefgr(g,f,F);[ 9, 30 ] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ T^2 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ], 3, 528, 1 ] Time: 1008.189528 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ T^2 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]*t^8 + [ T^12 + T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]*t^7 + [ 4*T^15 + T^14 + T^13 + 2*T^3 + T^2 + 5*T + 6, T^13 + T^12 + T^3 + 5*T^2 + 5*T + 6, T^3 + T^2 + 2, 3, 0, 0, 0, 0, 0, 0, 0 ]*t^6 + [ 4*T^17 + 4*T^16 + T^5 + T^4 + 5*T^3 + 4*T^2 + 6*T, 4*T^16 + 4*T^15 + 3*T^4 + 6*T^3 + 3*T + 6, T^5 + T^4 + 4*T^2 + 3*T + 6, T^4 + T^3 + T^2 + 2*T + 1, 3*T + 3, 0, 0, 0, 0, 0, 0 ]*t^5 + [ 4*T^27 + 4*T^16 + T^15 + 5*T^13 + 6*T^12 + T^4 + 5*T^2 + 6*T, 4*T^16 + 4*T^15 + 2*T^14 + 3*T^13 + 6*T^12 + T^4 + 3*T^3 + T^2 + 3*T + 6, T^15 + 5*T^13 + T^12 + T^4 + 2*T^3 + 3*T^2 + 3*T + 6, 3*T^12 + T^4 + T^3 + 5*T^2 + 2*T + 1, 3*T + 3, 0, 0, 0, 0, 0, 0 ]*t^4 + [ 4*T^29 + 4*T^28 + 4*T^18 + 5*T^17 + 4*T^15 + 4*T^14 + 6*T^13 + T^6 + T^5 + 3*T^4 + 2*T^3 + 6*T^2 + 5*T + 5, 4*T^28 + 4*T^27 + 4*T^18 + 5*T^17 + 4*T^16 + 2*T^15 + 3*T^13 + 6*T^12 + T^6 + 5*T^5 + 5*T^4 + 4*T^3 + 3*T^2 + 6*T + 6, 5*T^17 + T^16 + 4*T^15 + 4*T^14 + 3*T^13 + 6*T^12 + T^6 + 4*T^5 + 5*T^4 + T^3 + 4*T^2 + T + 4, T^16 + T^15 + T^14 + 2*T^13 + T^12 + T^6 + 3*T^5 + 2*T^4 + 3*T^2 + 4*T + 6, 3*T^13 + 3*T^12 + T^5 + 2*T^3 + 6*T^2 + 4*T + 5, 3*T^2 + 3, 0, 0, 0, 0, 0 ]*t^3 + [ 6*T^18 + 5*T^17 + 6*T^16 + 5*T^6 + 3*T^5 + 5*T^4 + 5*T^3 + 3*T^2 + 5*T,2*T^17 + T^16 + 6*T^15 + 2*T^4 + 3*T^3 + 5*T^2 + 2*T + 5, 6*T^18 + 6*T^17 + 3*T^16 + 3*T^15 + 3*T^6 + T^5 + 5*T^4 + 4*T^3 + 3*T^2 + 5*T + 6, 6*T^17 + 6*T^16 + 5*T^5 + 4*T^4 + 5*T^3 + 6*T^2 + 5*T + 5, 5*T^6 + 5*T^5 + 2*T^4 + T^3 + 4*T^2 + 5*T, 5*T^5 + 5*T^4 + T^3 + 6*T^2 + 2*T + 4, T^2 + T, 0, 0, 0, 0 ]*t^2 + [ 6*T^28 + 6*T^27 + 6*T^17 + 4*T^16 + 5*T^15 + 5*T^13 + 5*T^12 + 5*T^5 + 5*T^4 + 5*T^2 + 5*T, 3*T^27 + 6*T^17 + T^16 + T^15 + 4*T^14 + T^13 + 6*T^12 + 5*T^5 + 5*T^4 + 2*T^3 + 6*T^2 + 2*T + 5, 6*T^28 + 6*T^17 + 6*T^16 + T^15 + 5*T^14 + 5*T^12 + 3*T^5 + 4*T^3 + 5*T^2 + 5*T + 6, 6*T^17 + 6*T^16 + 2*T^15 + T^14 + T^13 + 3*T^5 + 3*T^4 + 4*T^3 + 6*T^2 + 5*T + 5, 5*T^16 + 5*T^13 + 4*T^12 + 5*T^5 + 2*T^4 + 3*T^3 + 5*T, T^13 + 5*T^5 + 5*T^4 + 6*T^2 + 2*T + 4, T^2 + T, 0, 0, 0, 0 ]*t + [ 6*T^30 + 5*T^29 + 6*T^28 + 6*T^19 + 3*T^18 + 2*T^17 + 5*T^16 + 5*T^15 + 3*T^14 + 5*T^13 + 5*T^7 + 3*T^6 + 5*T^5 + 5*T^4 + 3*T^3 + 5*T^2, 2*T^29 + T^28 + 6*T^27 + 6*T^19 + 6*T^18 + 5*T^17 + 3*T^15 + 5*T^14 + 2*T^13 + 5*T^12 + 5*T^7 + T^6 + 3*T^5 + 6*T^4 + 6*T^3 + 3*T^2 + 3*T, 6*T^30 + 6*T^29 + 3*T^28 + 3*T^27 + 6*T^19 + 4*T^18 + T^16 + 3*T^15 + 3*T^14 + 5*T^13 + 6*T^12 + 3*T^7 + T^6 + 2*T^4 + 4*T^3 + 5*T^2 + 6*T + 5, 6*T^29 + 6*T^28 + 6*T^19 + 4*T^18 + 6*T^17 + 3*T^16 + T^15 + 6*T^14 + 5*T^13 + 5*T^12 + 3*T^7 + 2*T^6 + 3*T^5 + 6*T^3 + 6*T^2 + 2*T + 6, 4*T^18 + 3*T^17 + T^16 + T^15 + 4*T^14 + 5*T^13 + 5*T^7 + 3*T^6 + 4*T^5 + 3*T^4 + T^3 + 2*T^2 + 3*T + 5, 5*T^17 + 5*T^16 + T^15 + 6*T^14 + 2*T^13 + 4*T^12 + 5*T^7 + T^6 + 5*T^5 + 4*T^4 + 4*T^3 + 4*T^2 + 2*T, T^14 + T^13 + 5*T^6 + 3*T^5 + 6*T^4 + T^3 + 2*T^2 + 6*T + 4, T^3 + 2*T^2 + T, 0, 0, 0 ] ####################################################### ####################################################### #### #### 13. BEISPIEL ####################################################### ####################################################### k := GF(13,4); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f:= y^7+(T^2+T)*y^6+(T^3+T^2)*y^5+(T^3+3*T^2+3*T+1)*y^4+(T^5+1)*y^3+(3*T+3)*y^2+(3*T+3)*y+T^11+T^9+T^6+T^3+T^2+1; F := FunctionField(f); Genus(F); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); s := Coerce(o,T); h1 := t^7+(6*s^7+13*s^6+s^4+8*s^3+9*s^2+13*s+5)*t^5+(10*s^18+10*s^17+8*s^6+13*s^5+16*s^3+3*s^2+6*s+6)*t^4+(8*s^13+10*s^12+8*s^11+12*s^10+4*s^9+7*s^8+3*s^7+2*s^6+11*s^5+11*s^4+12*s^3+9*s^2+4*s)*t^3+(9*s^15+6*s^14+6*s^13+14*s^12+4*s^11+11*s^10+s^9+8*s^8+s^7+12*s^6+3*s^5+13*s^4+16*s^2+14*s+5)*t^2+(11*s^17+6*s^6+4*s^15+5*s^14+15*s^13+8*s^12+9*s^11+14*s^10+3*s^9+6*s^8+9*s^7+2*s^6+2*s^5+7*s^4+15*s^3+16*s^2+13*s+7)*t+6*s^22+12*s^20+2*s^19+s^17+3*s^16+s^15+16*s^14+8*s^13+7*s^12+13*s^15+5*s^10+6*s^9+6*s^8+15*s^7+5*s^6+10*s^5+16*s^4+16*s^2+9*s+6;; h2 := t+Coerce(o,[11,11*T,7,0,0,0,0]); h3 := t+Coerce(o,[1,1,0,0,0,0,0]); g :=h1*h2*h3;; Coefgr(g,f,F);[9,23] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ************************************ [ [ t + [ 11, 11*T, 7, 0, 0, 0, 0 ] ], 4, 336, 1 ] Time: 1155.669138 s Das Polynom g hat folgende Gestalt: *********************************** t^9 + [ 12, 11*T + 1, 7, 0, 0, 0, 0 ]*t^8 + [ 6*T^7 + T^4 + 8*T^3 + 9*T^2 + 3, 11*T + 11, 11*T + 7, 7, 0, 0, 0 ]*t^7 + [ 10*T^18 + 10*T^17 + 7*T^7 + 8*T^6 + 12*T^4 + 8*T^3 + 7*T^2 + 6*T + 1, T^8 + 6*T^7 + 11*T^5 + 11*T^4 + 3*T^3 + 9*T^2 + 3*T + 5, 3*T^7 + 7*T^4 + 4*T^3 + 11*T^2 + 9, 0, 0, 0, 0 ]*t^6 + [ 3*T^18 + 3*T^17 + 8*T^13 + 10*T^12 + 8*T^11 + 12*T^10 + 4*T^9 + 7*T^8 + 4*T^7 + 7*T^6 + 11*T^5 + 9*T^4 + 6*T^3 + T^2 + 11*T + 10, 6*T^19 + 3*T^18 + 10*T^17 + T^8 + 11*T^7 + 8*T^6 + 11*T^5 + 2*T^4 + 2*T^3 + 12*T^2 + 10*T + 9, 5*T^18 + 5*T^17 + T^8 + 3*T^7 + 4*T^6 + 11*T^5 + 4*T^4 + 7*T^3 + 6*T^2 + 6*T + 12, 3*T^7 + 7*T^4 + 4*T^3 + 11*T^2 + 9, 0, 0, 0 ]*t^5 + [ 6*T^18 + 6*T^17 + 9*T^15 + 6*T^14 + 11*T^13 + 4*T^12 + 9*T^11 + 12*T^10 + 10*T^9 + T^8 + 11*T^7 + 7*T^6 + 5*T^5 + 2*T^4 + 8*T^3 + T^2 + 11*T + 6, 6*T^19 + 12*T^18 + 6*T^17 + 10*T^14 + T^13 + 7*T^12 + 10*T^11 + 4*T^10 + 3*T^9 + T^8 + 9*T^7 + 3*T^6 + 2*T^5 + 7*T^4 + 8*T^3 + 9*T^2 + 6*T + 1, 6*T^19 + 11*T^18 + 5*T^17 + 4*T^13 + 5*T^12 + 4*T^11 + 6*T^10 + 2*T^9 + 10*T^8 + 5*T^7 + 5*T^6 + 12*T^5 + 6*T^4 + 8*T^3 + 7*T^2 + 6*T + 3, 5*T^18 + 5*T^17 + 4*T^6 + 8*T^3 + 8*T^2 + 3*T + 3, 0, 0, 0 ]*t^4 + [ 11*T^17 + 8*T^15 + 12*T^14 + 6*T^13 + 2*T^11 + 5*T^10 + 7*T^9 + 10*T^8 + 2*T^7 + 5*T^6 + 3*T^5 + 11*T^4 + 4*T^3 + 8*T^2 + 4*T + 2, 8*T^16 + 10*T^15 + 4*T^14 + 7*T^13 + 9*T^12 + 7*T^11 + 3*T^10 + 2*T^9 + 12*T^8 + 6*T^7 + 6*T^6 + 11*T^5 + 6*T^4 + 4*T^3 + T^2 + 9*T + 5, 11*T^15 + 9*T^12 + 8*T^11 + 10*T^10 + 8*T^9 + 8*T^8 + 11*T^7 + 11*T^6 + 11*T^5 + T^4 + T^3 + 11*T^2 + 9*T + 9, 4*T^13 + 5*T^12 + 4*T^11 + 6*T^10 + 2*T^9 + 10*T^8 + 8*T^7 + T^6 + 12*T^5 + 12*T^4 + 6*T^3 + 11*T^2 + 2*T, 0, 0, 0 ]*t^3 + [ 6*T^22 + 12*T^20 + 2*T^19 + 3*T^17 + 3*T^16 + 5*T^15 + 12*T^14 + 7*T^13 + 10*T^12 + 9*T^11 + 8*T^10 + T^9 + 10*T^8 + 4*T^7 + 12*T^6 + 2*T^5 + 9*T^4 + 11*T^3 + 7*T^2 + 7*T + 2, 4*T^18 + 11*T^17 + 3*T^15 + 3*T^14 + 11*T^13 + 6*T^12 + 3*T^11 + 10*T^10 + 12*T^9 + 9*T^8 + 6*T^7 + 8*T^5 + 3*T^4 + 3*T^3 + 8*T^2 + 10, 12*T^17 + 8*T^16 + T^15 + 2*T^13 + 3*T^12 + 4*T^11 + 4*T^10 + 12*T^9 + 5*T^8 + 7*T^7 + 4*T^6 + 9*T^5 + 10*T^4 + 8*T^3 + T^2 + 10*T + 6, 11*T^15 + 3*T^14 + 3*T^13 + 7*T^12 + 2*T^11 + 12*T^10 + 7*T^9 + 4*T^8 + 7*T^7 + 6*T^6 + 8*T^5 + 8*T^2 + 7*T + 9, 0, 0, 0 ]*t^2 + [ 7*T^22 + T^20 + 11*T^19 + 3*T^17 + 10*T^16 + 4*T^15 + T^13 + 3*T^12 + 8*T^11 + 6*T^10 + T^9+ 8*T^8 + 6*T^7 + 5*T^6 + 12*T^5 + 9*T^4 + 9*T^3 + 4*T^2 + 4*T + 6, T^23 + 6*T^22 + 2*T^21 + 8*T^20 + 2*T^19 + 2*T^18 + 12*T^17 + 6*T^16 + 3*T^15 + 12*T^14 + 12*T^12 + 9*T^11 + 11*T^10 + 2*T^9 + 11*T^8 + 10*T^7 + 4*T^6 + 12*T^5 + 11*T^4 + 10*T^3 + 5*T^2 + 9*T + 5, 3*T^22 + 6*T^20 + T^19 + 4*T^18 + 6*T^17 + 12*T^15 + 2*T^13 + 9*T^12 + 9*T^11 + 10*T^10 + 12*T^9 + T^8 + 9*T^7 + 9*T^6 + 5*T^5 + T^4 + 8*T^3 + 3*T^2 + 10*T, 12*T^17 + 2*T^15 + 9*T^14 + T^13 + 4*T^12 + 11*T^11 + 7*T^10 + 8*T^9 + 3*T^8 + 11*T^7 + 4*T^6 + T^5 + 10*T^4 + T^3 + 8*T^2 +10, 0, 0, 0 ]*t + [ T^22 + 2*T^20 + 9*T^19 + 11*T^17 + 7*T^16 + 11*T^15 + 7*T^14 + 10*T^13 + 12*T^12 + 3*T^10 + T^9 + T^8 + 9*T^7 + 3*T^6 + 6*T^5 + 7*T^4 + 7*T^2 + 8*T + 1, T^23 + T^22 + 2*T^21 + 11*T^20 + 9*T^19 + 11*T^18 + 5*T^17 + 5*T^16 + 5*T^15 + 4*T^14 + 9*T^13 + 12*T^12 + 3*T^11 + 4*T^10 + 2*T^9 + 10*T^8 + 12*T^7 + 9*T^6 + 7*T^4 + 7*T^3 + 2*T^2 + 9*T + 1, T^23 + 3*T^22 + 2*T^21 + 2*T^20 + T^19 + 11*T^18 + T^17 + 6*T^16 + T^15 + 5*T^14 + 3*T^13 + 10*T^12 + 3*T^11 + 10*T^10 + 4*T^9 + 12*T^8 + 4*T^7 + 2*T^6 + 12*T^5 + 8*T^4 + 7*T^3 + 3*T^2 + 12*T + 3, 3*T^22 + 6*T^20 + T^19 + 7*T^17 + 8*T^16 + 7*T^15 + 8*T^14 + 4*T^13 + 10*T^12 + 9*T^10 + 3*T^9 + 3*T^8 + T^7 + 9*T^6 + 5*T^5 + 8*T^4 + 8*T^2 + 11*T + 3, 0, 0, 0 ] ####################################################### ####################################################### #### #### 14. BEISPIEL ####################################################### ####################################################### k := GF(7,5); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^5+2*y^2*T+2*y*T+T+1;; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 :=t+Coerce(o,[T+1,2*T,T,0,0]); h2 :=t-Coerce(o,[1,T,T,0,T]); h3 :=t+Coerce(o,[T^2+1,1,T^3,1,0]); h4 := t^3+Coerce(o,[3,6,T+1,T+1,1]); h5 := t^2+Coerce(o,[T^2+1,2*T+1,4*T,1,6+T]); g := h1*h2*h3*h4*h5;; Coefgr(g,f,F);[8,11]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ T + 1, 2*T, T, 0, 0 ] ], 3, 257, 1 ]---->>> Time: 131.171446 s Das Polynom g hat folgende Gestalt: *********************************** t^8 + [ T^2 + T + 1, T + 1, T^3, 1, 6*T ]*t^7 + [ 3*T^3 + 4*T^2 + T, T^5 + T^4 + 6*T^3 + 2*T^2 + T + 1, 2*T^5 + T^4 + 6*T^3 + 4*T, 2*T^5 + T^4 + 2*T^3 + 6*T^2 + T + 1, 6*T^3 + 6 ]*t^6 + [ 5*T^7 + T^6 + 5*T^5 + T^4 + T^3 + 4*T^2 + T + 4, 3*T^7 + T^6 + 2*T^4 + T^3 + 6*T^2 + T + 6, 3*T^7 + 5*T^6 + 2*T^5 + 4*T^4 + 3*T^3 + 6*T^2 + 5*T + 3, 2*T^6 + 2*T^5 + T^4 + 2*T^3 + 4*T^2 + 2*T + 2, 6*T^6 + 4*T^5 + 5*T^3 + 2*T^2 + 2 ]*t^5 + [ 5*T^7 + 4*T^6 + 5*T^5 + 6*T^4 + 5*T^3 +3*T^2 + 6*T, 2*T^7 + 3*T^6 + 3*T^5 + 3*T^4 + 4*T^3 + 5*T^2 + T, 6*T^7 + 5*T^6 + 6*T^5 + 6*T + 6, T^7 + 2*T^6 + 2*T^4 + 2*T^3 + 6*T^2 + 2*T + 4, 3*T^7 + T^5 + 5*T^4 + 5*T^3 + 2*T^2 + 2*T + 2 ]*t^4 + [ 6*T^8 + 4*T^6 + T^5 + 3*T^4 + 4*T^3 + 5*T^2 + T + 2, 4*T^9 + 2*T^8 + 4*T^7 + 2*T^6 + 3*T^5 + 3*T^4 + 6*T^3 + 5*T^2 + 6*T + 2, T^9 + 4*T^8 + 6*T^7 + 3*T^6 + 2*T^3 + T^2 + 4, T^9 + 5*T^7 + 4*T^6 + 3*T^5 + 3*T^4 + 6*T^3 + 3*T^2 + 4*T + 3, T^7 + T^6 + T^5 + 6*T^4 + 3*T^3 + 6*T^2 + 4*T + 3 ]*t^3 + [ 4*T^9 + 3*T^8 + 2*T^7 + T^5 + T^4 + 5*T^3 + T + 1, T^9 + T^8 + 6*T^7 + 6*T^6 + 5*T^5 + 5*T^4 + 5*T^3 + T^2 + T + 2, T^9 + 3*T^8 + 2*T^7 + 2*T^6 + 4*T^3 + 2*T^2 + 4*T + 5, 2*T^8 + 2*T^7 + 3*T^6 + 5*T^4 + 5*T^3 + 5*T^2 + 3*T + 1, T^8 + 2*T^7 + 5*T^6 + 4*T^5 + 3*T^4 + 5*T^2 + T + 5 ]*t^2 + [ 6*T^9 + 5*T^6 + 5*T^5 + 5*T^4 + 6*T^3 + 6*T^2 + 4*T, T^9 + 5*T^7 + 5*T^6 + 6*T^5 + 4*T^4 + 6*T^3 + 5*T^2 + T + 5, T^9 + 4*T^8 + 5*T^7 + 5*T^6 + 6*T^5 + T^4 + T^3 + T^2 + T, 2*T^6 + 3*T^5 + 5*T^4 + 4*T^3 + 3*T^2 + 4*T + 1, T^9 + T^8 + 2*T^7 + 4*T^6 + 4*T^5 + T^2 + 3*T + 2 ]*t + [ 5*T^11 + 2*T^10 + T^9 + 3*T^8 + 2*T^7 + 2*T^6 + 6*T^5 + 2*T^4 + 6*T^3 + T + 5, 2*T^11 + 5*T^10 + 4*T^9 + 6*T^8 + 2*T^7 + 6*T^5 + T^4 + 4*T^3 + 6*T^2 + 4*T + 4, T^11 + 3*T^10 + 4*T^9 + T^8 + 6*T^7 + T^6 + 3*T^5 + 3*T^4 + 3*T^2 + 3, 5*T^11 + 5*T^10 + 5*T^9 + T^8 + 3*T^6 + 6*T^4 + 2*T^2 + 3*T + 1, 3*T^10 + T^9 + 4*T^8 + 2*T^7 + 2*T^6 + T^5+ T^4 + 2*T^3 + 2*T^2 + 4*T + 1 ] ####################################################### ####################################################### #### #### 15. BEISPIEL ####################################################### ####################################################### k :=GF(17,3); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f:= y^7+(T^2+T)*y^6+(T^3+T^2)*y^5+(T^3+3*T^2+3*T+1)*y^4+(T^5+1)*y^3+(3*T+3)*y^2+ (3*T+3)*y+T^11+T^9+T^6+T^3+T^2+1; F := FunctionField(f); Genus(F); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; s := Coerce(o,T); h1 := t^7+(11*s^7+s^4+4*s^3+10)*t^6+(6*s^11+13*s^6+s^4+8*s^3+9*s^2+13*s+5)*t^5+(10*s^8+10*s^7+8*s^6+13*s^5+11*s^3+3*s^2+6*s+6)*t^4+(s^13+10*s^12+8*s^11+12*s^10+4*s^9+7*s^8+3*s^7+2*s^6+11*s^5+11*s^4+12*s^3+9*s^2+4*s)*t^3+(2*s^15+6*s^14+6*s^13+14*s^12+4*s^11+11*s^10+s^9+8*s^8+s^7+12*s^6+3*s^5+13*s^4+16*s^2+14*s+5)*t^2+(s^17+6*s^6+4*s^15+5*s^14+15*s^13+8*s^12+9*s^11+14*s^10+3*s^9+6*s^8+9*s^7+2*s^6+2*s^5+7*s^4+15*s^3+16*s^2+13*s+7)*t+6*s^24+12*s^22+2*s^19+s^17+3*s^16+s^15+16*s^14+8*s^11+7*s^13+s^11+5*s^10+6*s^9+6*s^8+15*s^7+5*s^6+10*s^5+16*s^4+16*s^2+9*s+6; h2 := ot.1 + Coerce(o,[ 16, 16*T, 14, 0, 0, 0,0]); g := h1*h2;; Coefgr(g,f,F);[8,25] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ ot.1 + [ 16, 16*T, 14, 0, 0, 0, 0 ] ], 4, 343,1 ] Time: 174.910867 s Das Polynom g hat folgende Gestalt: *********************************** ot.1^8 + [ 11*T^7 + T^4 + 4*T^3 + 9, 16*T, 14, 0, 0, 0, 0 ]*ot.1^7 + [ 6*T^11 + 6*T^7 + 13*T^6 + 4*T^3 + 9*T^2 + 13*T + 12, 6*T^8 + 16*T^5 + 13*T^4 + 7*T, T^7 + 14*T^4 + 5*T^3 + 4, 0, 0, 0, 0 ]*ot.1^6 + [ 11*T^11 + 10*T^8 + 10*T^7 + 12*T^6 + 13*T^5 + 16*T^4 + 3*T^3 + 11*T^2 + 10*T + 1, 11*T^12 + 4*T^7 + 16*T^5 + 9*T^4 + 8*T^3 + 4*T^2 + 12*T, 16*T^11 + 12*T^6 + 14*T^4 + 10*T^3 + 7*T^2 + 12*T + 2, 0, 0, 0, 0 ]*ot.1^5 + [ T^13 + 10*T^12 + 8*T^11 + 12*T^10 + 4*T^9 + 14*T^8 + 10*T^7 + 11*T^6 + 15*T^5 + 11*T^4 + T^3 + 6*T^2 + 15*T + 11, 7*T^9 + 7*T^8 + 9*T^7 + 4*T^6 + 6*T^4 + 14*T^3 + 11*T^2 + 11*T, 4*T^8 + 4*T^7 + 10*T^6 + 12*T^5 + T^3 + 8*T^2 + 16*T + 16, 0, 0, 0, 0 ]*ot.1^4 + [ 2*T^15 + 6*T^14 + 5*T^13 + 4*T^12 + 13*T^11 + 16*T^10 + 14*T^9 + T^8 + 15*T^7 + 10*T^6 + 9*T^5 + 2*T^4 + 5*T^3 + 7*T^2 + 10*T + 5, 16*T^14 + 7*T^13 + 9*T^12 + 5*T^11 + 13*T^10 + 10*T^9 + 14*T^8 + 15*T^7 + 6*T^6 + 6*T^5 + 5*T^4 + 8*T^3 + 13*T^2, 14*T^13 + 4*T^12 + 10*T^11 + 15*T^10 + 5*T^9 + 13*T^8 + 8*T^7 + 11*T^6 + T^5 + T^4 + 15*T^3 + 7*T^2 + 5*T, 0, 0, 0, 0 ]*ot.1^3 + [ T^17 + 2*T^15 + 16*T^14 + 9*T^13 + 11*T^12 + 5*T^11 + 3*T^10 + 2*T^9 + 15*T^8 + 8*T^7 + 13*T^6 + 16*T^5 + 11*T^4 + 15*T^3 + 16*T + 2, 15*T^16 + 11*T^15 + 11*T^14 + 3*T^13 + 13*T^12 + 6*T^11 + 16*T^10 + 9*T^9 + 16*T^8 + 5*T^7 + 14*T^6 + 4*T^5 + T^3 + 3*T^2 + 12*T, 11*T^15 + 16*T^14 + 16*T^13 + 9*T^12 + 5*T^11 + T^10 + 14*T^9 + 10*T^8 + 14*T^7 + 15*T^6 + 8*T^5 + 12*T^4 + 3*T^2 + 9*T + 2, 0, 0, 0, 0 ]*ot.1^2 + [ 6*T^24 + 12*T^22 + 2*T^19 + 3*T^16 + 14*T^15 + 11*T^14 + 9*T^13 + 9*T^12 + 8*T^10 + 3*T^9 + 6*T^7 + 14*T^6 + 8*T^5 + 9*T^4 + 2*T^3 + 13*T + 16, 16*T^18 + 13*T^16 + 12*T^15 + 2*T^14 + 9*T^13 + 8*T^12 + 3*T^11 + 14*T^10 + 11*T^9 + 8*T^8 + 9*T^7 + 15*T^6 + 10*T^5 + 2*T^4 + T^3 + 4*T^2 + 10*T, 14*T^17 + 5*T^15 + 2*T^14 + 6*T^13 + 10*T^12 + 7*T^11 + 9*T^10 + 8*T^9 + 16*T^8 + 7*T^7 + 10*T^6 + 11*T^5 + 13*T^4 + 6*T^3 + 3*T^2 + 12*T + 13, 0, 0, 0, 0 ]*ot.1 + [ 11*T^24 + 5*T^22 + 15*T^19 + 16*T^17 + 14*T^16 + 16*T^15 + T^14 + 10*T^13 + 8*T^11 + 12*T^10 + 11*T^9 + 11*T^8 + 2*T^7 + 12*T^6 + 7*T^5 + T^4 + T^2 + 8*T + 11, 11*T^25 + 5*T^23 + 15*T^20 + 16*T^18 + 14*T^17 + 16*T^16 + T^15 + 10*T^14 + 8*T^12 + 12*T^11 + 11*T^10 + 11*T^9 + 2*T^8 + 12*T^7 + 7*T^6 + T^5 + T^3 + 8*T^2 + 11*T, 16*T^24 + 15*T^22 + 11*T^19 + 14*T^17 + 8*T^16 + 14*T^15 + 3*T^14 + 13*T^13 + 7*T^11 + 2*T^10 + 16*T^9 + 16*T^8 + 6*T^7 + 2*T^6 + 4*T^5 + 3*T^4 + 3*T^2 + 7*T + 16, 0, 0, 0, 0 ] ####################################################### ####################################################### #### #### 16. BEISPIEL ###################################################### ####################################################### k := GF(29,3); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^11+11*y^5*T^3+3*y^3*T^7+4*T*y^2+T^11+T^7+12*T^2+7*T+4; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[3*T^33+1,T^6,12*T^11,0,0,0,0,0,0,0,0]); h2 := t-Coerce(o,[2*T^3+2*T^2+T+2,1,2*T^3+1,12*T+13,0,0,0,0,0,0,0]); h3 := t^2+Coerce(o,[2*T^3+T^2+5,3*T^3+2*T^2+1,T+1,0,0,0,0,0,0,0,0]); h4 := t+Coerce(o,[2,13*T+T^2+1,1,0,0,0,0,0,0,0,0]); h5 := t+Coerce(o,[12,15*T+3*T^2+1,1,0,0,0,0,0,0,0,0]); h6 := t+Coerce(o,[11,15,T,0,0,0,0,0,0,0,0]); g := h1*h2*h3*h4*h5*h6;; Coefgr(g,f,F); [8,47]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); ************************************ [ [ t + [ 11, 15, T, 0, 0, 0, 0, 0, 0, 0, 0 ] ], 4, 1271, 1 ] Time: 2159.442164 s Das Polynom g hat folgende Gestalt: *********************************** t^8 + [27*T^3 + 27*T^2 + 28*T + 23, 4*T^2 + 28*T + 16, 27*T^3 +T+1,17*T + 16, 0,0,0,0,0,0,0]*t^7 + [ 3*T^33 + 10*T^3 + 9*T^2 + 4*T + 18, T^6 + 21*T^5 + 23*T^4 + 25*T^3 + 23*T^2 + 15*T + 14, 12*T^11 + T^4 + 27*T^3 + 18*T^2+ 25*T + 22, 21*T^5 + 2*T^4 + 28*T^3 + 28*T^2 + 20*T + 7, 27*T^4 + 6*T^3 + 18*T^2 + 13*T + 10, 17*T^2 + 21*T + 3, 0, 0, 0, 0, 0 ]*t^6 + [ 23*T^36 + 23*T^35 + 26*T^34 + 11*T^33 + 25*T^6 + 23*T^5 + 25*T^4 + 25*T^3 + 3*T^2 + 19*T + 17, 12*T^35 + 26*T^34 + 19*T^33 + 27*T^9 + 27*T^8 + 28*T^7 + 17*T^6 + 19*T^5 + 14*T^4 + 4*T^3 + 13*T^2 + 22*T + 21, 23*T^36 + 3*T^34 + 3*T^33 + 5*T^14 + 5*T^13 + 17*T^12 + 15*T^11 + 4*T^8 + 22*T^7 + 14*T^6 + 19*T^5 + 15*T^4 + 6*T^3 + 12*T^2 + 11*T + 7, 22*T^34 + 19*T^33 + 19*T^13 + 17*T^12 + 18*T^11 + 27*T^9 + T^7 + 16*T^6 + 3*T^5 + 7*T^4 + 2*T^2 + 25*T + 18, 5*T^14 + 12*T^12 + 12*T^11 + 11*T^7 + 24*T^6 + 7*T^5 + 13*T^4 + 4*T^3 + 23*T^2 + 8*T + 7, T^12 + 18*T^11 + 21*T^6 + 16*T^5 + 7*T^4 + 12*T^3 + 5*T^2 + 8*T + 26, 6*T^4 + 26*T^3 + 7*T^2 + 8*T + 18, 5*T^2 + 20*T + 16, 0, 0, 0 ]*t^5 + [ T^36 + 27*T^35 + 12*T^34 + 22*T^33 + 16*T^6 + 24*T^5 + 16*T^4 + 14*T^3 + 26*T^2 + 21*T + 13, 5*T^38 + 11*T^37 + 17*T^36 + 11*T^35 + 16*T^34 + 13*T^33 + 10*T^9 + 22*T^8 + 13*T^7 + 21*T^6 + 8*T^5 + 21*T^4 + 14*T^3 + 9*T^2 + 12*T + 17, 3*T^37 + 23*T^36 + 25*T^35 + 17*T^34 + 8*T^33 + 4*T^14 + 21*T^13 + 19*T^12 + 22*T^11 + 23*T^10 + 25*T^9 + 28*T^8 + 4*T^7 + 26*T^5 + 26*T^4 + 12*T^3 + 16*T^2 + 8*T + 16, 5*T^38 + 6*T^37 + 26*T^36 + 26*T^35 + 2*T^34 + 21*T^33 + 20*T^16 + 15*T^15 + 10*T^14 + 15*T^13 + 6*T^12 + 23*T^11 + T^10 + 27*T^9 + 2*T^8 + 21*T^7 + 4*T^6 + 13*T^5 + 8*T^4 + 3*T^3 + 11*T^2 + 23*T + 17, 23*T^37 + 18*T^36 + 25*T^35 + 10*T^34 + T^33 + 12*T^15 + 5*T^14 + 13*T^13 + 10*T^12 + 24*T^11 + 2*T^10 + 28*T^9 + 27*T^8 + 3*T^6 + 17*T^5 + 14*T^4 + 17*T^3 + T^2 + 12*T + 10, 22*T^35 + 5*T^34 + 9*T^33 + 20*T^16 + 24*T^15 + 17*T^14 + 17*T^13 + 8*T^12 + 26*T^11 + 27*T^10 + 6*T^9 + 18*T^8 + 4*T^7 + 5*T^6 + 16*T^5 + 6*T^4 + 10*T^3 + 14*T^2 + 23*T + 4, 5*T^15 + 14*T^14 + 13*T^13 + 11*T^12 + 4*T^11 + 11*T^8 + 11*T^6 + 26*T^5 + 22*T^4 + 5*T^3 + 17*T^2 + 16*T + 23, T^13 + 20*T^12 + 7*T^11 + 14*T^6 + 11*T^5 + 9*T^4 + 23*T^3 + 26*T^2 + 22*T + 6, 8*T^4 + 18*T^3 + 18*T^2 + 25*T +8,17*T^2+16*T,0]*t^4 + [ 17*T^39 + 11*T^38 + 17*T^37 + 23*T^36 + 15*T^35 + 2*T^34 + 11*T^33 + 9*T^6 + 28*T^5 + 9*T^4 + 2*T^3 + 18*T^2 + 27, 11*T^39 + 28*T^38 + 13*T^37 + 12*T^36 + 27*T^35 + 11*T^34 + 15*T^33 + 25*T^12 + 23*T^11 + 25*T^10 + 27*T^9 + 18*T^8 + 21*T^7 + 2*T^6 + 24*T^5 + 10*T^4 + 13*T^3 + 2*T^2 + 19*T + 3, 11*T^40 + 23*T^39 + 28*T^38 + 16*T^37 + 24*T^36 + 7*T^35 + T^34 + 18*T^33 + 10*T^17 + 15*T^16 + 10*T^15 + 5*T^14 + 2*T^13 + 2*T^12 + 5*T^11 + 2*T^10 + 2*T^9 + 5*T^8 + 28*T^7 + 21*T^6 + 23*T^5 + 25*T^4 + 17*T^3 + 20*T^2 + 8*T + 3, 16*T^39 + 9*T^38 + 21*T^37 + 6*T^35 + 24*T^34 + 6*T^33 + 15*T^17 + 25*T^16 + 23*T^15 + 19*T^14 + 15*T^13 + 13*T^12 + 21*T^11 + 26*T^10 + 15*T^9 + 20*T^8 + 28*T^7 + 4*T^6 + 18*T^5 + 19*T^4 + 20*T^2 + 9*T, 11*T^40 + 24*T^39 + 21*T^38 + 10*T^37 + 12*T^36 + 11*T^35 + 24*T^34 + 21*T^33 + 15*T^18 + 5*T^17 + 25*T^16 + 6*T^15 + 9*T^14 + 28*T^13 + 19*T^12 + 17*T^11 + 24*T^10 + 15*T^9 + 23*T^8 + T^7 + 14*T^6 + 6*T^5 + 17*T^4 + 3*T^3 + 6*T^2 + 9*T + 27, 5*T^39 + 19*T^38 + 21*T^37 + 7*T^36 +15*T^35 + 24*T^34 + 20*T^33 + 6*T^17 + 7*T^16 + 26*T^15 + 18*T^13 + 17*T^12 + 2*T^11 + 24*T^10 + 26*T^8 + 17*T^7 + 16*T^6 + 2*T^5 + 8*T^4 + 9*T^3 + 16*T^2 + 3*T, 18*T^37 + 20*T^36 + 21*T^35 + 24*T^34 + 25*T^33 + 15*T^18 + 9*T^17 + 26*T^16 + 11*T^15 + 19*T^14 + 15*T^13 + T^12 + 13*T^11 + 7*T^10 + 17*T^9 + 2*T^8 + 17*T^7 + 25*T^6 + 15*T^5 + 19*T^3 + 18*T^2 + 2*T + 21, 15*T^35 + 2*T^34 + 19*T^33 + 20*T^17 + 18*T^16 + 26*T^15 + 28*T^14 + 2*T^13 + 9*T^12 + 22*T^11 + 6*T^10 + 26*T^9 + 7*T^8 + 18*T^7 + 19*T^5 + 21*T^4 + 3*T^3 + 23*T^2 + 26*T + 8, 14*T^15 + 22*T^14 + 26*T^13 + 9*T^12 + 13*T^11 + 5*T^8 + 20*T^7 + 16*T^6 + 21*T^5+15*T^4+5*T^3+23*T^2+17*T+5,2*T^13+8*T^12+18*T^11+5*T^3+25*T^2+7*T+16,0 ]*t^3 + [ 19*T^39 + 14*T^38 + 19*T^37 + 12*T^36 + 22*T^35 + 22*T^34 + 17*T^33 + 28*T^24 + 11*T^23 + 28*T^20 + 11*T^19 + 17*T^15 + 21*T^14 + 11*T^13 + 28*T^12 + 12*T^10 + 25*T^9 + 13*T^8 + 4*T^6 + 5*T^5 + 11*T^4 + 19*T^3 + 27*T^2 + 25*T + 24, 10*T^41 + 27*T^40 + 12*T^39 + 19*T^38 + 23*T^37 + 25*T^36 + 16*T^35 + 20*T^34 + 9*T^33 + 16*T^12 + 24*T^11 + 16*T^10 + 4*T^9 + 21*T^8 + 21*T^7 + 25*T^6 + 16*T^5 + 3*T^4 + 22*T^3 + 2*T^2 + 19*T + 4, 15*T^41 + 25*T^40 + 16*T^39 + 20*T^38 + 17*T^37 + 13*T^36 + 23*T^35 + 7*T^34 + 11*T^33 + 18*T^17 + 27*T^16 + 18*T^15 + 28*T^14 + 25*T^13 + 5*T^12 + 26*T^11 + 11*T^10 + 25*T^9 + 3*T^8 + 16*T^7 + 24*T^6 + 2*T^5 + 28*T^4 + 28*T^3 + 9*T^2 + 9*T + 17, 10*T^41 + 17*T^40 + 4*T^39 + 5*T^38 + 18*T^37 + 12*T^36 + 7*T^35 + 9*T^34 + T^33 + 26*T^20 + 15*T^19 + 21*T^18 + 19*T^17 + 18*T^16 + 5*T^15 + 18*T^14 + 24*T^13 + 8*T^12 + 4*T^11 + 2*T^10 + 8*T^9 + 26*T^8 + 15*T^7 + 15*T^6 + 10*T^5 + 28*T^4 + 17*T^3 + 22*T^2 + 7*T + 27, 26*T^41 + 27*T^40 + 17*T^39 + 22*T^38 + 19*T^37 + 4*T^36 + 7*T^35 + 26*T^34 + 2*T^19 + 13*T^18 + 6*T^17 + 22*T^16 + 10*T^15 + 7*T^14 + T^13 + 10*T^12 + 24*T^11 + 8*T^10 + 2*T^9 + 4*T^8 + 16*T^7 + 3*T^6 + 17*T^5 + 11*T^4 + 25*T^3 + 9*T^2 + 22*T + 24, 2*T^40 + 14*T^39 + 19*T^38 + 18*T^37 + T^36 + 20*T^35 + 6*T^34 + 3*T^33 + 11*T^19 + 10*T^18 + 16*T^17 + 9*T^16 + 19*T^15 + 18*T^14 + 8*T^13 + 3*T^12 + 3*T^11 + 9*T^10 + 16*T^9 + 28*T^8 + 9*T^7 + 14*T^6 + 24*T^5 + 9*T^4 + 12*T^3 + 26*T^2 + 21*T + 28, 11*T^41 + 24*T^40 + 24*T^39 + 20*T^38 + 8*T^37 + 15*T^36 + 22*T^35 + 19*T^34 + 11*T^33 + 17*T^19 + 21*T^18 + 10*T^17 + T^16 + 18*T^15 + 16*T^14 + 19*T^13 + 12*T^12 + 4*T^11 + 7*T^10 + 18*T^9 + 3*T^8 + 4*T^7 + 2*T^6 + 16*T^5 + 14*T^4 + 22*T^3 + 25*T^2 + 17*T + 22, 13*T^39 + 4*T^38 + 27*T^37 + 11*T^36 + 20*T^35 + 8*T^34 + 18*T^33 + 8*T^18 + 27*T^17 + 18*T^16 + 14*T^15 + 27*T^14 + T^13 + 3*T^12 + 20*T^11 + 5*T^10 + 15*T^9 + 20*T^8 + 17*T^7 + 27*T^6 + 4*T^5 + 3*T^4 + 26*T^3 + 9*T^2 + 13*T, 24*T^37 + 25*T^36 + 25*T^35 + 17*T^34 + 24*T^33 + 15*T^19 + 9*T^18 + 9*T^17 + 22*T^16 + 3*T^15 + 2*T^14 + T^13 + 3*T^12 + 26*T^11 + 9*T^10 + T^9 + 2*T^8 + 16*T^7 + 27*T^6 + 26*T^5 + 4*T^4 + 24*T^3 + 16*T^2 + 14*T +3, 22*T^35 + 19*T^34 + 23*T^17 + 16*T^16 + 21*T^15 + 15*T^14 + 22*T^13 + 3*T^12 + 14*T^11 + 8*T^10 + 18*T^9 + 18*T^8 + 5*T^7 + 16*T^6 + 28*T^5 + 9*T^4 + 15*T^3 + 6*T^2 + 6*T + 11, 9*T^15 + 13*T^14 + 13*T^13 + 10*T^12 + 9*T^11 + 17*T^8 + 16*T^7 + T^5 + 21*T^4 + 10*T^3 + 2*T^2 + 20*T + 8 ]*t^2 + [ 10*T^39 + 15*T^38 + 10*T^37 + 12*T^36 + 10*T^35 + 27*T^34 + 12*T^33 + 27*T^25 + 19*T^24 + 3*T^23 + 11*T^22 + 27*T^21 + 19*T^20 + 3*T^19 + 11*T^18 + 5*T^16 + 11*T^15 + 16*T^14 + 26*T^13 + 2*T^12 + 15*T^11 + 13*T^6 + 5*T^5 + 13*T^4 + 4*T^3 + 13*T^2 + 9*T + 4, 10*T^41 + 3*T^40 + 13*T^39 + 15*T^38 + 17*T^37 + 27*T^36 + 8*T^35 + 17*T^34 + 23*T^33 + 13*T^12 + 5*T^11 + 13*T^10 + 4*T^9 + 26*T^8 + 10*T^7 + 18*T^6 + 5*T^5 + 25*T^4 + 9*T^3 + 22*T^2 + 25*T + 27, 22*T^43 + 23*T^42 + 17*T^41 + 4*T^40 + 25*T^39 + 12*T^38 + T^37 + 27*T^36 + 24*T^35 + 23*T^34 + 20*T^33 + 11*T^17 + 2*T^16 + 3*T^15 + 21*T^14 + 24*T^13 + 21*T^12 + 24*T^11 + 13*T^10 + 7*T^9 + 18*T^8 + 7*T^7 + 16*T^6 + 4*T^5 + 10*T^4 + 9*T^3 + 8*T^2 + 27*T + 26,4*T^43 + 21*T^42 + 24*T^41 + 25*T^40 + 7*T^39 + 16*T^38 + 7*T^37 + 25*T^35 + 3*T^34 + 23*T^33 + 23*T^21 + 28*T^20 + 20*T^19 + 16*T^18 + 23*T^17 + 19*T^16 + 8*T^15 + 17*T^14 + 14*T^13 + 28*T^12 + 9*T^11 + 21*T^10 + 16*T^9 + 16*T^8 + 16*T^7 + 9*T^6 + 15*T^5 + 12*T^4 + 18*T^2 + T + 27, 22*T^43 + 16*T^42 + 5*T^41 + 26*T^40 + 12*T^39 + 26*T^38 + 12*T^37 + 26*T^36 + 7*T^35 + 3*T^34 + 2*T^33 + T^21 + 5*T^20 + 10*T^19 + 16*T^18 + 13*T^17 + T^16 + 11*T^15 + 27*T^13 + 17*T^12 + 8*T^11 + 15*T^9 + 10*T^8 + 2*T^6 + 28*T^5 + 4*T^4 + 28*T^3 + 12*T^2 + T + 20, 4*T^43 + 17*T^42 + 9*T^41 + 27*T^40 + 22*T^39 + 16*T^38 + 3*T^37 + 20*T^36 + 4*T^35 + 14*T^34 + 9*T^33 + 16*T^21 + 26*T^20 + 9*T^19 + 13*T^18 + 6*T^17 + 18*T^15 + 26*T^14 + 12*T^13 + 16*T^12 + 4*T^11 + 15*T^10 + 24*T^9 + 15*T^8 + 10*T^7 + 8*T^6 + 15*T^5 + T^4 + 26*T^3 + 11*T^2 + 24*T + 3, 15*T^42 + 20*T^41 + 27*T^40 + 26*T^39 + 16*T^38 + 11*T^37 + 8*T^36 + 4*T^35 + 11*T^34 + 9*T^33 + T^21 + 6*T^20 + 20*T^19 + 17*T^18 + 19*T^17 + 28*T^16 + 15*T^15 + 20*T^14 + 8*T^13 + 23*T^11 + T^10 + 2*T^9 + 8*T^8 + 4*T^7 + 2*T^6 + 15*T^5 + 23*T^4 + 22*T^3 + 11*T^2 + 23*T + 3, T^40 + 4*T^39 + 28*T^38 + 5*T^37 + 9*T^36 + 25*T^35 + 18*T^34 + 5*T^33 + 16*T^21 + 10*T^20 + 7*T^19 + 21*T^18 + T^17 + 6*T^16 + 17*T^15 + 19*T^14 + 25*T^13 + 26*T^12 + 22*T^11 + 23*T^10 + 22*T^9 + 11*T^8 + 4*T^7 + 14*T^6 + 19*T^5 + 21*T^4 + 3*T^3 + 18*T^2 + 6*T + 21, 5*T^38 + 16*T^37 + 15*T^36 + 11*T^35 + 22*T^34 + 15*T^33 + 2*T^20 + 22*T^19 + 21*T^18 + 17*T^17 + 6*T^16 + 15*T^15 + 3*T^14 + 26*T^13 + 26*T^12 + 26*T^11 + 21*T^10 + 3*T^9 + 18*T^8 + 6*T^7 + 21*T^6 + 21*T^5 + 15*T^4 + 5*T^3 + 23*T^2 + 17*T + 5, 15*T^36 + 17*T^35 + 21*T^34 + 19*T^33 + 4*T^18 + 16*T^17 + 25*T^16 + 20*T^15 + 7*T^14 + 13*T^13 + 14*T^12 + 12*T^11 + 15*T^10 + 5*T^9 + 23*T^8 + 17*T^7 + 5*T^6 + 5*T^3 + 25*T^2 + 7*T + 16, 20*T^16 + 6*T^15 + 2*T^14 + 15*T^13 + T^12 + 2*T^11 + 5*T^9 + 25*T^8 + 7*T^7 + 16*T^6 ]*t + [ 7*T^47 + 17*T^46 + 10*T^45 + 7*T^43 + 17*T^42 + 10*T^41 + 22*T^39 + T^38 + 14*T^37 + 16*T^36 + T^35 + 24*T^34 + 26*T^33 + 8*T^29 + 20*T^28 + 12*T^27 + 8*T^26 + 2*T^25 + 7*T^24 + 16*T^23 + 20*T^22 + 2*T^21 + 15*T^20 + 8*T^19 + 18*T^18 + 28*T^17 + 22*T^16 + 6*T^15 + 12*T^14 + 21*T^13 + 6*T^12 + 13*T^11 + 8*T^10 + 21*T^9 + T^8 + 9*T^7 + 14*T^6 + 10*T^5 + 24*T^4 + 15*T^3 + 10*T^2 + 8*T + 28, 2*T^41 + 14*T^40 + 17*T^39 + 15*T^37 + 12*T^36 + 19*T^35 + 8*T^34 + 3*T^33 + 17*T^27 + 9*T^26 + 25*T^25 + 5*T^24 + 9*T^23 + 25*T^21 + 17*T^20 + 17*T^19 + 5*T^18 + 24*T^17 + 8*T^16 + 6*T^15 + 23*T^14 + T^13 + 9*T^12 + 3*T^11 + 24*T^10 + 15*T^9 + T^8 + 3*T^7 + 24*T^6 + 5*T^4 + 4*T^3 + 16*T^2 + 22*T + 1, 10*T^43 + 21*T^42 + 7*T^41 + 3*T^40 + 18*T^39 + 15*T^38 + 9*T^37 + 13*T^36 + 4*T^35 + 20*T^34 + 11*T^33 + 28*T^25 + 10*T^24 + 11*T^23 + 28*T^21 + 10*T^20 + 14*T^19 + 22*T^18 + 20*T^17 + 7*T^16 + 3*T^15 + 3*T^14 + 15*T^13 + 24*T^12 + 20*T^11 + 7*T^10 + 3*T^9 + 6*T^8 + 20*T^7 + 7*T^6 + 5*T^5 + 3*T^4 + 14*T^3 + 11*T^2 + 26*T + 23, 18*T^43 + 11*T^42 + 16*T^41 + 21*T^40 + T^39 + 25*T^38 + 8*T^37 + 10*T^36 + T^35 + 12*T^34 + 22*T^33 + 24*T^25 + 2*T^24 + 7*T^23 + 24*T^22 + 11*T^21 + 19*T^20 + 20*T^19 + 5*T^18 + 15*T^17 + 19*T^16 + 16*T^15 + 20*T^14 + 21*T^13 + 2*T^12 + 17*T^11 + 9*T^10 + 8*T^9 + 26*T^8 + 4*T^7 + 4*T^6 + 18*T^5 + 22*T^4 + 13*T^3 + 10*T^2 + 4*T + 17, 22*T^44 + 6*T^43 + 23*T^42 + 8*T^41 + 12*T^40 + 20*T^39 + 14*T^37 + 13*T^36 + 20*T^35 + 11*T^34 + 14*T^33 + 22*T^23 + 27*T^22 + 28*T^21 + 12*T^20 + 4*T^19 + 14*T^18 + 14*T^17 + 8*T^16 + T^15 + 9*T^14 + 23*T^13 + 3*T^12 + 21*T^11 + 24*T^10 + 11*T^9 + 3*T^8 + 8*T^7 + 14*T^6 + 24*T^4 + 14*T^3 + 26*T^2 + 23*T + 24, 4*T^44 + 8*T^43 + 15*T^42 + 19*T^41 + 28*T^40 + 28*T^39 + 24*T^38 + 9*T^37 + 6*T^36 + 28*T^34 + 23*T^33 + 15*T^21 + 3*T^20 + 22*T^19 + 27*T^18 + 13*T^17 + 17*T^16 + 16*T^15 + 20*T^14 + 9*T^13 + 22*T^12 + 19*T^11 + 18*T^9 + 13*T^8 + 13*T^7 + 14*T^6 + 8*T^5 + 3*T^4 + 2*T^3 + 19*T + 27, 22*T^44 + 3*T^43 + 24*T^42 + 7*T^41 + 11*T^40 + 8*T^39 + 28*T^38 + 5*T^37 + 22*T^36 + 24*T^35 + 3*T^34 + 26*T^33 + T^22 + 24*T^21 + 5*T^20 + 16*T^19 + 2*T^18 + 18*T^17 + 19*T^16 + 2*T^15 + 27*T^14 + 12*T^13 + 5*T^12 + 23*T^11 + 4*T^10 + 10*T^9 + 12*T^8 + 13*T^7 + 20*T^6 + 19*T^5 + 21*T^4 + 17*T^3 + 8*T^2 + T + 28, 4*T^44 + 7*T^43 + T^42 + 9*T^41 + 3*T^40 + 28*T^39 + 8*T^38 + 11*T^37 + 9*T^36 + 7*T^35 + 2*T^34 + 11*T^33 + 16*T^22 + 3*T^21 + 2*T^20 + 18*T^19 + 25*T^18 + 13*T^17 + 10*T^16 + 15*T^15 + 7*T^14 + 23*T^13 + 18*T^12 + 6*T^11 + 4*T^10 + 27*T^9 + 11*T^8 + 2*T^7 + 18*T^6 + 22*T^5 + 23*T^4 + 3*T^3 + 12*T^2 + 20*T + 23, 21*T^42 + 15*T^41 +11*T^40 + 5*T^39 + 20*T^38 + 17*T^37 + 18*T^36 + 23*T^35 + 25*T^34 + 14*T^33 + T^22 + 12*T^21 + 9*T^20 + 28*T^19 + 15*T^18 + 14*T^17 + 8*T^16 + T^15 + 4*T^14 + 10*T^13 + 2*T^12 + 10*T^11 + 23*T^10 + 10*T^9 + 17*T^8 + 14*T^7 + 15*T^6 + 26*T^5 + 25*T^4 + 6*T^3 + 27*T^2 + 18*T + 24, 27*T^40 + 24*T^39 + 26*T^38 + 27*T^37 + 16*T^36 + 25*T^35 + 28*T^34 + 4*T^33 + 16*T^22 + 28*T^21 + 4*T^20 + 7*T^19 + 12*T^18 + 25*T^17 + 3*T^16 + 22*T^15 + 12*T^14 + 22*T^13 + 12*T^11 + 25*T^10 + 6*T^9 + 27*T^8 + 27*T^7 + 3*T^6 + 28*T^5 + 9*T^4 + 15*T^3 + 18*T^2 + 19*T + 11, 3*T^38 + 5*T^37 + T^36 + 6*T^35 + 2*T^34 + 24*T^33 + 26*T^20 + 2*T^19 + 15*T^18 + 20*T^17 + 22*T^16 + 10*T^15 + 14*T^14 + 14*T^13 + 21*T^12 + 26*T^11 + 9*T^10 + 15*T^9 + 18*T^8 + 19*T^7 + 11*T^6 + T^5 + 21*T^4 + 10*T^3 + 2*T^2 + 20*T + 8 ] ####################################################### ####################################################### #### #### 17. BEISPIEL ####################################################### ####################################################### k := GF(13,4); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f:= y^7+(T^2+T)*y^6+(T^3+T^2)*y^5+(T^3+3*T^2+3*T+1)*y^4+(T^5+1)*y^3+(3*T+3)*y^2+(3*T+3)*y+T^11+T^9+T^6+T^3+T^2+1; F := FunctionField(f); Genus(F); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); s := Coerce(o,T); h1 := t^7+(6*s^7+13*s^6+s^4+8*s^3+9*s^2+13*s+5)*t^5+(10*s^18+10*s^17+8*s^6+13*s^5+16*s^3+3*s^2+6*s+6)*t^4+(8*s^13+10*s^12+8*s^11+12*s^10+4*s^9+7*s^8+3*s^7+2*s^6+11*s^5+11*s^4+12*s^3+9*s^2+4*s)*t^3+(9*s^15+6*s^14+6*s^13+14*s^12+4*s^11+11*s^10+s^9+8*s^8+s^7+12*s^6+3*s^5+13*s^4+16*s^2+14*s+5)*t^2+(11*s^17+6*s^6+4*s^15+5*s^14+15*s^13+8*s^12+9*s^11+14*s^10+3*s^9+6*s^8+9*s^7+2*s^6+2*s^5+7*s^4+15*s^3+16*s^2+13*s+7)*t+6*s^22+12*s^20+2*s^19+s^17+3*s^16+s^15+16*s^14+8*s^13+7*s^12+13*s^15+5*s^10+6*s^9+6*s^8+15*s^7+5*s^6+10*s^5+16*s^4+16*s^2+9*s+6;; h2 := t+Coerce(o,[11,11*T,7,0,0,0,0]); g :=h1*h2;; Coefgr(g,f,F); [8,23] Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ 11, 11*T, 7, 0, 0, 0, 0 ] ], 4, 294, 1 ]--->>> Time: 306.505095 s Time: 405.848646 s Das Polynom g hat folgende Gestalt: *********************************** t^8 + [ 11, 11*T, 7, 0, 0, 0, 0 ]*t^7 + [ 6*T^7 + T^4 + 8*T^3 + 9*T^2 + 5, 0, 0, 0, 0, 0, 0 ]*t^6 + [ 10*T^18 + 10*T^17 + T^7 + 8*T^6 + 11*T^4 + 11*T^2 + 6*T + 9, T^8 + 11*T^5 + 10*T^4 + 8*T^3 + 3*T, 3*T^7 + 7*T^4 + 4*T^3 + 11*T^2 + 9, 0, 0, 0, 0 ]*t^5 + [ 6*T^18 + 6*T^17 + 8*T^13 + 10*T^12 + 8*T^11 + 12*T^10 + 4*T^9 + 7*T^8 + 3*T^7 + 12*T^6 + 11*T^5 + 11*T^4 + 6*T^3 + 3*T^2 + 5*T + 1, 6*T^19 + 6*T^18 + 10*T^7 + 7*T^4 + 7*T^3 + T^2 + T, 5*T^18 + 5*T^17 + 4*T^6 + 8*T^3 + 8*T^2 + 3*T + 3, 0, 0, 0, 0 ]*t^4 + [ 9*T^15 + 6*T^14 + 3*T^13 + 7*T^12 + T^11 + 6*T^9 + 7*T^8 + 8*T^7 + 8*T^6 + 7*T^5 + 4*T^4 + 2*T^3 + 11*T^2 + 6*T + 5, 10*T^14 + 6*T^13 + 10*T^12 + 2*T^11 + 5*T^10 + 12*T^9 + 7*T^8 + 9*T^7 + 4*T^6 + 4*T^5 + 2*T^4 + 8*T^3 + 5*T^2, 4*T^13 + 5*T^12 + 4*T^11 + 6*T^10 + 2*T^9 + 10*T^8 + 8*T^7 + T^6 + 12*T^5 + 12*T^4 + 6*T^3 + 11*T^2 + 2*T, 0, 0, 0, 0 ]*t^3 + [ 11*T^17 + 12*T^15 + 6*T^14 + 3*T^13 + 6*T^12 + T^11 + 5*T^10 + T^9 + 3*T^8 + 7*T^7 + 10*T^6 + 9*T^5 + 7*T^4 + 2*T^3 + 10*T^2 + 11*T + 10, 8*T^16 + T^15 + T^14 + 11*T^13 + 5*T^12 + 4*T^11 + 11*T^10 + 10*T^9 + 11*T^8 + 2*T^7 + 7*T^6 + 7*T^3 + 11*T^2 + 3*T, 11*T^15 + 3*T^14 + 3*T^13 + 7*T^12 + 2*T^11 + 12*T^10 + 7*T^9 + 4*T^8 + 7*T^7 + 6*T^6 + 8*T^5 + 8*T^2 + 7*T + 9, 0, 0, 0, 0 ]*t^2 + [ 6*T^22 + 12*T^20 + 2*T^19 + 5*T^17 + 3*T^16 + 6*T^15 + 6*T^14 + 4*T^13 + 4*T^12 + 8*T^11 + 3*T^10 + 7*T^8 + 10*T^7 + 2*T^6 + 6*T^5 + 2*T^4 + 9*T^3 + 10*T^2 + 9*T + 5, 4*T^18 + 5*T^16 + 3*T^15 + 9*T^14 + 10*T^13 + 8*T^12 + 11*T^11 + 7*T^10 + T^9 + 8*T^8 + 10*T^7 + 9*T^6 + 12*T^5 + 9*T^4 + 7*T^3 + 12*T, 12*T^17 + 2*T^15 + 9*T^14 + T^13 + 4*T^12 + 11*T^11 + 7*T^10 + 8*T^9 + 3*T^8 + 11*T^7 + 4*T^6 + T^5 + 10*T^4 + T^3 + 8*T^2 + 10, 0, 0, 0, 0 ]*t + [ T^22 + 2*T^20 + 9*T^19 + 11*T^17 + 7*T^16 + 11*T^15 + 7*T^14 + 10*T^13 + 12*T^12 + 3*T^10 + T^9 + T^8 + 9*T^7 + 3*T^6 + 6*T^5 + 7*T^4 + 7*T^2 + 8*T + 1, T^23 + 2*T^21 + 9*T^20 + 11*T^18 + 7*T^17 + 11*T^16 + 7*T^15 + 10*T^14 + 12*T^13 + 3*T^11 + T^10 + T^9 + 9*T^8 + 3*T^7 + 6*T^6 + 7*T^5 + 7*T^3 + 8*T^2 + T, 3*T^22 + 6*T^20 + T^19 + 7*T^17 + 8*T^16 + 7*T^15 + 8*T^14 + 4*T^13 + 10*T^12 + 9*T^10 +3*T^9 + 3*T^8 + T^7 + 9*T^6 + 5*T^5 + 8*T^4 + 8*T^2 + 11*T + 3, 0, 0, 0, 0 ] ###################################################### ###################################################### #### #### 18. BEISPIEL ####################################################### ####################################################### k := GF(19,2); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^7+2*T*y^6+3*y^3*T^2+T^11+T^3+2*T^2+15; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; h1 := t+Coerce(o,[14*T,T,1,0,T,0,0]); h2 :=t-Coerce(o,[T,17*T,1,0,13,0,0]); h3 :=t+Coerce(o,[T+1,T,14+T,0,0,2,0]); h4 := t+ Coerce(o,[3,15,0,0,0,0,0]); h5 := t^3+Coerce(o,[1,1,1,1,1,1,1]); g := h1*h2*h3*h4*h5;; Coefgr(g,f,F);[7,29]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); Fu"r Die Stelle( (T + k.1^197, F.1 + k.1^199)) [ [ ot.1 + [ 3, 15, 0, 0, 0, 0, 0 ] ], 2, 330, 1 ] Time: 59.159226 s Das Polynom g hat folgende Gestalt: *********************************** ot.1^7 + [ 14*T + 4, 4*T + 15, T + 14, 0, T + 6, 2, 0 ]*ot.1^6 + [ 11*T^14 + 2*T^13 + 11*T^6 + 5*T^5 + 4*T^4 + 13*T^3 + 10*T^2 + 17*T + 3, 4*T^13 + 18*T^12 + 4*T^5 + 7*T^4 + 17*T^3 + 8*T^2 + T + 15, 17*T^12 + 7*T^11 + 17*T^4 + 3*T^3 + 13*T^2 + 10*T + 14, 14*T^5 + 6*T^4 + 3*T^2 +T+1,12*T^4+16*T^3+18*T+4,13*T^3+5*T^2+15*T+1,3*T^4+4*T^3+T^2+6*T+6]*ot.1^5 + [ 11*T^18 + 12*T^16 + 2*T^15 + 6*T^14 + 17*T^13 + 18*T^12 + 18*T^11 + 11*T^10 + 3*T^9 + 12*T^8 + T^7 + 10*T^6 + 6*T^4 + 16*T^3 + 8*T^2 + 5*T + 5, 4*T^17 + 9*T^15 + 18*T^14 + 12*T^13 + 12*T^12 + 11*T^11 + 4*T^9 + 8*T^8 + 9*T^7 + T^6 + 10*T^5 + 14*T^3 + T^2 + 4*T + 14, 17*T^16 + 7*T^14 + 10*T^13 + T^12 + 4*T^11 + 17*T^8 + 15*T^7 + 7*T^6 + 13*T^5 + 2*T^4 + 12*T^3 + 2*T^2 + 14*T + 4, T^15 + 5*T^13+ 12*T^12 + 10*T^11 + 14*T^9 + 18*T^7 + 8*T^6 + 4*T^5 + 12*T^4 + 3*T^3 + 13*T^2 + 3*T + 18, 9*T^14 + 2*T^12 + 7*T^11 + 12*T^8 + 17*T^6 + 15*T^5 + 12*T^3 + 18*T^2 + 4*T + 6, 5*T^13 + 13*T^7 + 7*T^5 + 2*T^4 + 4*T^3 + 9*T^2 + 16*T, 7*T^12 + 3*T^8 +8*T^6+4*T^5+15*T^4+9*T^3+15*T^2+9*T+16]*ot.1^4 + [ 9*T^23 + 12*T^19 + 14*T^18 + 13*T^17 + 13*T^16 + 3*T^15 + 18*T^14 + 11*T^13 + 13*T^12 + 3*T^9 + 10*T^8 + 2*T^7 + 12*T^6 + 16*T^5 + 6*T^3 + 4*T^2 + 8*T + 14, 13*T^18 + 12*T^17 + 9*T^16 + 10*T^15 + 9*T^14 + 4*T^13 + 7*T^12 + 11*T^11 + 13*T^10 + 14*T^8 + 14*T^7 + 5*T^5 + 13*T^4 + 8*T^3 + 10*T^2 + 9*T + 13, 3*T^17 + 13*T^16 + 2*T^15 + T^14 + 2*T^13 + 2*T^12 + 11*T^11 + 3*T^9 + 9*T^7 + 12*T^6 + 9*T^5 + 17*T^4 + 4*T^3 + T^2 + 4*T + 8, 8*T^16 + 3*T^15 + 18*T^14 + 6*T^13 + 3*T^12 + 3*T^11 + 17*T^10 + 4*T^9 + 9*T^8 + T^7 + 17*T^6 + 13*T^5 + 17*T^4 + 10*T^3 + 12*T^2 + 13*T + 2, 15*T^15 + 8*T^14 + 18*T^13 + 9*T^12 + T^9 + 17*T^8 + 4*T^7 + 11*T^6 + 4*T^5 + 16*T^4 + 8*T^3 + 12*T^2 + 17*T + 17, 2*T^14 + 15*T^13 + 11*T^12 + 10*T^11 + 9*T^8 + T^7 + 8*T^6 + 3*T^5 + 9*T^4 + 10*T^3 + 9*T^2 + 9*T + 5, 17*T^13 + 2*T^12 + 5*T^9 + 9*T^8 + 12*T^7 + 16*T^6 + 17*T^5 + 18*T^4+15*T^3+10*T^2+13*T+17]*ot.1^3 + [ 11*T^24 + 17*T^23 + 12*T^22 + T^20 + 4*T^19 + 16*T^18 + 18*T^17 + 13*T^16 + 5*T^15 + 5*T^14 + 6*T^13 + 10*T^12 + 12*T^11 + 5*T^10 + 8*T^9 + 3*T^8 + T^7 + 9*T^6 + 12*T^5 + 18*T^4 + 11*T^3 + 3*T^2 + 13*T + 15, 2*T^23 + 12*T^22 + 9*T^19 + 17*T^18 + 2*T^17 + 12*T^16 + 5*T^15 + 3*T^14 + 5*T^13 + 3*T^12 + 11*T^11 + 16*T^10 + 17*T^9 + 18*T^8 + 16*T^7 + 4*T^6 + 2*T^5 + 18*T^4 + 10*T^3 + 9*T^2 + 8*T + 8, 5*T^18 + T^17 + 13*T^16 + 12*T^15 + 11*T^14 + 9*T^13 + 17*T^12 + 14*T^11 + 5*T^10 + 11*T^9 + 15*T^8 + 18*T^7 + 12*T^6 + 17*T^5 + 6*T^4 + 4*T^3 + 14*T^2 + 5*T + 4, 7*T^17 + 9*T^16 + 10*T^15 + 17*T^14 + 16*T^13 + 10*T^12 + 18*T^11 + 12*T^10 + 17*T^9 + T^8 + T^7 + 18*T^6 + 7*T^4 + 3*T^3 + 12*T^2 + 15*T + 1, 6*T^16 + 5*T^15 + 2*T^14 + 10*T^13 + 7*T^12 + 8*T^10 + 13*T^9 + 12*T^8 + 15*T^7 + 15*T^6 + 17*T^5 + 11*T^4 + 6*T^3 + 13*T^2 + 7*T + 8, 16*T^15 + 7*T^14 + 5*T^13 + 10*T^12 + T^11 + 15*T^9 + 3*T^8 + 17*T^7 + 18*T^6 + 14*T^5 + 2*T^4 + 6*T^3 + 14*T^2 + 10*T + 5, 14*T^14 + 2*T^13 + 2*T^12 + 7*T^11 + 2*T^10 + 8*T^9 + 15*T^8 + 6*T^7 + 3*T^6 + 10*T^5 + T^3 + 9*T^2 +5*T+6]*ot.1^2 + [ 14*T^28 + 10*T^27 + 4*T^26 + 18*T^25 + 2*T^24 + 17*T^23 + 6*T^22 + 5*T^21 + 3*T^20 + 6*T^19 + 11*T^18 + 3*T^17 + 11*T^16 + 17*T^15 + 10*T^14 + 15*T^13 + 15*T^12 + 2*T^11 + 17*T^10 + 13*T^9 + 17*T^8 + 14*T^7 + 10*T^6 + 10*T^4 + 9*T^3 + 5*T^2 + 7*T + 3, 10*T^27 + 15*T^26 + 8*T^25 + 13*T^24 + 2*T^23 + 3*T^22 + 10*T^21 + 2*T^20 + 2*T^19 + 4*T^18 + 17*T^17 + 10*T^16 + 12*T^15 + 16*T^14 + 7*T^13 + 13*T^12 + 12*T^11 + 8*T^10 + 4*T^9 + 11*T^8 + T^7 + 4*T^6 + 3*T^5 + 4*T^4 + 13*T^3 + 8*T^2 + 2, 15*T^26 + 11*T^25 + 8*T^24 + 12*T^23 + 7*T^22 + 7*T^21 + 9*T^20 + 11*T^19 + 7*T^18 + 11*T^17 + 17*T^16 + 14*T^15 + 18*T^14 + 13*T^13 + 12*T^12 + 17*T^11 + 5*T^10 + 13*T^9 + 16*T^8 + 14*T^7 + 15*T^6 + 5*T^5 + 15*T^4 + 18*T^3 + 18*T^2 + 18*T + 12, 11*T^25 + 9*T^24 + 10*T^23 + 12*T^22 + 7*T^21 + 6*T^20 + 2*T^19 + 9*T^18 + 2*T^17 + 10*T^16 + 14*T^15 + 5*T^13 + 18*T^12 + 10*T^11 + 3*T^10 + 7*T^9 + 9*T^8 + 10*T^7 + 3*T^6 + 17*T^5 + 10*T^4 + 8*T^3 + 11*T + 18, 9*T^24 + 12*T^23 + 6*T^20 + 16*T^19 + 6*T^18 + 11*T^17 + 5*T^16 + 6*T^15 + 11*T^14 + 11*T^13 + 13*T^11 + 3*T^10 + 10*T^9 + 5*T^8 + 2*T^7 + 9*T^6 + 7*T^5 + 13*T^4 + 6*T^3 + 16*T^2 + 10*T + 3, 12*T^23 + 16*T^19 + 11*T^18 + 3*T^17 + T^16 + 14*T^15 + 15*T^14 + 17*T^13 + 9*T^12 + 2*T^11 + 13*T^9 + 15*T^8 + 2*T^7 + 8*T^6 + 13*T^5 + 17*T^4 + 6*T^3 + 11*T^2 + 6*T + 6, T^18 + 5*T^17 + 13*T^16 + 18*T^15 + 17*T^14 + 5*T^13 + 9*T^12 + 2*T^11 + 2*T^10 + 13*T^9 + 5*T^8 + 14*T^6 + 15*T^5 + 10*T^4 + 7*T^3 + 8*T^2 + 8*T + 18 ]*ot.1 + [ 4*T^29 + 5*T^28 + 6*T^27 + 6*T^26 + 5*T^25 + 9*T^24 + 12*T^23 + 11*T^22 + 8*T^21 + 9*T^20 + 8*T^19 + 8*T^18 + 5*T^17 + 4*T^16 + 12*T^15 + 9*T^14 + 11*T^13 + 4*T^12 + 3*T^11 + 12*T^10 + 17*T^9 + 12*T^6 + 11*T^5 + 9*T^4 + 8*T^3 + 17*T + 14, T^28 + 9*T^27 + 10*T^26 + 5*T^25 + 4*T^24 + 18*T^23 + 11*T^22 + 4*T^21 + 13*T^20 + 9*T^19 + 10*T^18 + 10*T^17 + T^16 + 8*T^15 + 15*T^14 + 12*T^13 + 17*T^12 + 18*T^11 + 9*T^8 + 7*T^7 + 4*T^4 + 13*T^3 + 15*T^2 + 9*T + 11, 17*T^27 + 4*T^26 + T^25 + 5*T^24 + 13*T^23 + 10*T^22 + 3*T^20 + 6*T^19 + 17*T^18 + T^17 + 18*T^16 + 8*T^15 + 9*T^14 + 7*T^12 + 14*T^11 + 14*T^10 + 14*T^9 + 12*T^8 + 15*T^7 + 11*T^6 + 18*T^5 + 14*T^4 + 5*T^3 + 10*T^2 + 16*T + 11, 16*T^26 + 8*T^25 + 14*T^24 + 7*T^23 + 6*T^22 + 12*T^21 + 13*T^20 + 11*T^19 + 3*T^18 + 6*T^17 + 6*T^16 + 8*T^15 + 7*T^14 + 8*T^13 + 17*T^12 + 7*T^11 + 3*T^10 + 14*T^9 + 11*T^8 + 7*T^7 + T^6 + 16*T^5 + 7*T^4 + 4*T^3 + 5*T + 10, 13*T^25 + 10*T^24 + 15*T^23 + 9*T^22 + 10*T^21 + 13*T^20 + 2*T^19 + 16*T^18 + 6*T^17 + 12*T^16 + 4*T^15 + 12*T^14 + 16*T^13 + 14*T^12 + 4*T^11 + 17*T^10 + 6*T^9 + 10*T^8 + 11*T^7 + 4*T^6 + 7*T^4 + 16*T^3 + 12*T^2 + 11*T + 5, 2*T^24 + 7*T^23 + 14*T^20 + 3*T^19 + 9*T^18 + 5*T^17 + 2*T^16 + 5*T^15 + 18*T^14 + 7*T^13 + 6*T^12 + 12*T^11 + 7*T^10 + 18*T^9 + 4*T^8 + 17*T^7 + 11*T^6 + 10*T^5 + 17*T^4 + 13*T^3 + 16*T^2 + 4*T + 18, 9*T^23 + T^19 + 18*T^18 + 12*T^17 + 3*T^16 + 14*T^15 + 15*T^14 + 3*T^13 + 16*T^12 + 9*T^11 + 12*T^10 + 11*T^9 + 2*T^8 + T^7 + 9*T^6 + 8*T^5 + 18*T^4 + 14*T^3 + 17*T^2 + 12*T + 14 ] ####################################################### ####################################################### #### #### 19. BEISPIEL ####################################################### ####################################################### k := GF(19,2); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^11+11*y^5*T^3+3*y^3*T^7+4*T*y^2+T^11+T^7+12*T^2+7*T+4; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[2*T^12+3*T+12,2*T^2+1,0,0,0,0,0,0,0,0,0]); h2 := t+Coerce(o,[T,T+1,0,0,0,0,0,0,0,0,0]); h3 := t-Coerce(o,[T^4+2,T^7,2,0,0,0,0,0,0,0,0]); h4 := t^2+Coerce(o,[2*T^11+3*T^2+11,7*T^3+14,0,0,0,0,0,0,0,0,0]); h5:= t+Coerce(o,[T,T,1,1,0,0,0,0,0,0,0]); g := h1*h2*h3*h4*h5;; Coefgr(g,f,F);[7,22]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); *********************************** [ [ t + [ T, T + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ], 3, 528, 1 ] Time: 114.680869 s Das Polynom g hat folgende Gestalt: *********************************** t^7 + [ 18*T^4 + 2*T + 17, 18*T^7 + 2*T + 1, 18, 1, 0, 0, 0, 0, 0, 0, 0 ]*t^6 + [ 2*T^12 + 17*T^5 + T^2 + 14, 17*T^8 + 17*T^5 + 18*T^4 + 4*T^2 + 17*T + 18, 17*T^8 + 18*T^7 + 18*T^4 + T^2 + 17*T, 18*T^7 + 18*T^4 + 17*T + 17, 18*T^7 + T + 18, 17, 0, 0, 0, 0, 0 ]*t^5 + [ 17*T^16 + 4*T^13 + 15*T^12 + 18*T^6 + 15*T^5 + 5*T^4 + 6*T^2 + T + 10, 17*T^19 + 4*T^13 + 2*T^12 + 18*T^9 + 15*T^8 + 5*T^7 + 15*T^6 + 17*T^5 + 18*T^4 + 4*T^3 + 2*T^2 + 11*T + 12, 17*T^12 + 15*T^9 + 17*T^8 + 18*T^7 + 18*T^6 + 17*T^5 + 17*T^4 + 4*T^3 + 18*T + 2, 2*T^12 + 18*T^9 + 17*T^8 + 17*T^7 + 17*T^5 + 17*T^4 + 13*T^2 + 3*T + 11, 17*T^8 + 17*T^7 + 18*T^5 + 18*T^4 + 16*T + 17, 18*T^8 + 18*T^7 + 15*T + 18, 17*T + 18, 0, 0, 0, 0 ]*t^4 + [ 15*T^17 + 2*T^14 + 13*T^13 + 4*T^12 + 11*T^6 + 10*T^5 + 4*T^3 + T^2 + 5, 15*T^20 + 15*T^17 + 17*T^16 + 4*T^14 + 15*T^13 + 15*T^12 + 11*T^9 + 10*T^8 + 15*T^7 + 9*T^6 + 4*T^5 + 7*T^4 + 3*T^3 + T^2 + 16*T + 12, 15*T^20 + 17*T^19 + 17*T^16 + 2*T^14 + 15*T^13 + 15*T^10 + 9*T^9 + 4*T^8 + T^7 + 15*T^6 + 8*T^5 + 8*T^4 + 2*T^3 + 3*T^2 + 15*T + 13, 17*T^19 + 17*T^16 + 15*T^13 + 15*T^12 + 15*T^10 + 15*T^9 + 8*T^8 + 4*T^7 + 17*T^6 + 8*T^5 + 4*T^4 + 16*T^3 + 15*T^2 + 6*T + 4, 17*T^19 + 2*T^13 + 17*T^12 + 17*T^9 + 8*T^8 + 2*T^7 + 17*T^6 + 16*T^5 + 15*T^4 + 15*T^3 + 2*T^2 + 17*T + 16, 15*T^12 + 17*T^9 + 16*T^8 + 15*T^7 + 16*T^4 + 2*T^3 + 5*T + 1, 16*T^7 + 18*T^4 +15*T^2+17*T+12,18*T^7+T+14,17,0,0]*t^3 + [ 17*T^18 + 17*T^17 + 15*T^16 + 4*T^13 + 11*T^12 + 15*T^7 + 2*T^6 + T^5 + 14*T^4 + 17*T^3 + 2*T^2 + 12*T + 9, 17*T^21 + 17*T^20 + 15*T^19 + 15*T^18 + 15*T^17 + 2*T^13 + 4*T^12 + 15*T^10 + 2*T^9 + 18*T^8 + 3*T^7 + 17*T^6 + 11*T^5 + 17*T^4 + 17*T^3 + 4*T^2 + 16*T + 1, 15*T^21 + 15*T^20 + 17*T^18 + 15*T^17 + 15*T^16 + 15*T^14 + 7*T^12 + 17*T^11 + 8*T^10 + 17*T^9 + 7*T^8 + 7*T^7 + 11*T^6 + 2*T^5 + 14*T^4 + 7*T^3 + 13*T + 6, 17*T^21 + 15*T^20 + 15*T^19 + 15*T^17 + 15*T^16 + 11*T^14 + 15*T^11 + 9*T^10 + 11*T^9 + 9*T^7 + 18*T^6 + 18*T^5 + 6*T^4 + 18*T^3 + 5*T^2 + T + 9, 15*T^20 + 15*T^19 + 17*T^17 + 17*T^16 + 15*T^14 + 13*T^13 + 13*T^12 + 17*T^11 + 14*T^10 + 18*T^9 + 18*T^8 + 6*T^7 + 5*T^6 + 12*T^5 + 14*T^4 + 7*T^3 + 11*T^2 + 5*T + 17, 17*T^20 + 17*T^19 + 11*T^13 + 17*T^12 + 15*T^10 + 5*T^9 + 12*T^8 + T^7 + 15*T^6 + 11*T^5 + 12*T^4 + 11*T^3 + 15*T^2 + 13*T + 15, 15*T^13 + 17*T^12 + 17*T^10 + 15*T^9 + 11*T^8 + 16*T^7 + 15*T^5 + 16*T^4 + 11*T^3 + 16*T^2 + 15, 15*T^8 + 16*T^7 + 18*T^5 + 18*T^4 + 15*T^3 + 13*T^2 + T + 12, 18*T^8 + 18*T^7 + 11*T + 13, 17*T + 17, 0 ]*t^2 + [ 15*T^18 + 11*T^17 + 2*T^15 + 15*T^14 + 3*T^13 + 13*T^7 + 2*T^6 + 9*T^5 + 3*T^4 + 6*T^3 + 9*T^2 + 18*T, 15*T^21 + 11*T^20 + 11*T^18 + 9*T^17 + 15*T^16 + 6*T^15 + 13*T^14 + 3*T^13 + 11*T^12 + 13*T^10 + 2*T^9 + 5*T^8 + 18*T^7 + 11*T^6 + 8*T^5 + 12*T^3 + 14*T^2 + 17*T + 9, 11*T^21 + 9*T^20 + 15*T^19 + 15*T^18 + 7*T^17 + 15*T^16 + 6*T^15 + 17*T^14 + 6*T^13 + 11*T^12 + 15*T^11 + 18*T^10 + 11*T^9 + 17*T^8 + 17*T^7 + 7*T^6 + 18*T^5 + 15*T^4 + T^3 + 5*T^2 + 16*T + 5, 15*T^21 + 7*T^20 + 15*T^19 + 3*T^17 + 11*T^16 + 2*T^15 + 2*T^14 + 18*T^12 + 11*T^11 + 3*T^10 + 7*T^9 + 8*T^8 + 8*T^6 + 6*T^5 + 8*T^4 + 5*T^3 + 3*T^2 + 11*T + 9, 3*T^20 + 11*T^19 + 13*T^17 + 13*T^16 + 6*T^14 + T^13 + 3*T^12 + 15*T^11+ 7*T^10 + 8*T^9 + 10*T^7 + 2*T^6 + 6*T^5 + 10*T^3 + 9*T^2 + T + 8, 13*T^20 + 13*T^19 + 13*T^16 + 4*T^14 + T^13 + 14*T^12 + 3*T^10 + 2*T^9 + 4*T^8 + 11*T^7 + 13*T^6 + 7*T^5 + 5*T^4 + 7*T^3 + 3*T^2 + 18*T, 13*T^19 + 17*T^16 + 15*T^13 + 9*T^12 + 13*T^10 + 13*T^9 + 7*T^8 + 18*T^7 + 13*T^6 + 16*T^5 + 8*T^4 + T^3 + 10*T^2 + 9*T + 4, 17*T^19 + 2*T^13 + 9*T^12 + 13*T^9 + 16*T^8 + 4*T^7 + 17*T^6 + 18*T^4 + 15*T^3 + 12*T^2 + 14*T + 11, 15*T^12 + 17*T^9 + 18*T^7 + 2*T^3 + 9*T^2 + 14*T+9,15*T^2+17,0]*t + [ 17*T^19 + 15*T^18 + 15*T^15 + 11*T^14 + 16*T^8 + T^7 + 14*T^6 + 13*T^4 + 2*T^3 + 9*T^2, 17*T^22 + 15*T^21 + 13*T^19 + 9*T^18 + 15*T^17 + 7*T^15 + 18*T^14 + 11*T^13 + 16*T^11 + T^10 + 12*T^9 + 6*T^8 + 5*T^7 + 8*T^6 + 10*T^5 + 12*T^4 + 10*T^3 + 16*T^2 + 9*T, 13*T^22 + 9*T^21 + 15*T^20 + 13*T^19 + 5*T^18 + 11*T^17 + 3*T^15 + 2*T^14 + 3*T^13 + 17*T^12 + 6*T^11 + 5*T^10 + 2*T^9 + 14*T^8 + 12*T^7 + 13*T^6 + 14*T^5 + 13*T^4 + 7*T^3 + 16*T^2 + 14*T, 13*T^22 + 5*T^21 + 11*T^20 + 17*T^19 + T^18 + 5*T^17 + 15*T^16 + 3*T^15 + T^14 + 2*T^13 + 5*T^12 + 12*T^10 + 7*T^9 + 9*T^8 + 7*T^7 + 16*T^6 + 4*T^5 + 11*T^4 + 5*T^3 + 10*T^2 + 11*T + 9, 17*T^22 + T^21 + 5*T^20 + 15*T^19 + 5*T^18 + T^17 + 15*T^16 + 7*T^15 + T^14 + 5*T^13 + 5*T^12 + 2*T^11 + 7*T^10 + 14*T^9 + 2*T^8 + 16*T^7 + 9*T^6 + 15*T^5 + 14*T^4 + 9*T^3 + 6*T^2 + 5, 5*T^21 + T^20 + 15*T^19 + 15*T^18 + 5*T^17 + 15*T^16 + 15*T^15 + 13*T^14 + T^13 + T^12 + T^11 + 2*T^10 + 9*T^9 + 17*T^8 + 7*T^7 + T^6 + 3*T^5 + 7*T^4 + 4*T^3 + 15*T^2 + 13*T + 14, 15*T^21 + 5*T^20 + 15*T^19 + 11*T^17 + 13*T^16 + 10*T^14 + 5*T^13 + 18*T^12 + 5*T^11 + 14*T^10 + T^9 + 11*T^8 + 17*T^7 + T^6 + 8*T^5 + 13*T^4 + 14*T^3 + T^2 + 10*T + 5, 11*T^20 + 13*T^19 + 17*T^17 + 17*T^16 + 11*T^14 + 6*T^13 + 7*T^12 + 15*T^11 + 5*T^10 + T^9 + 12*T^8 + 11*T^7 + 10*T^6 + 14*T^4 + 12*T^3 + 3*T^2 + 11*T + 13, 17*T^20 + 17*T^19 + 3*T^13 + 7*T^12 + 11*T^10 + 10*T^9 + 2*T^7 + 17*T^6 + 18*T^5 + 10*T^4 + 6*T^3 + 17*T^2 + 3*T + 17, 15*T^13 + 15*T^12 + 17*T^10 + 17*T^9 + 18*T^8 + 18*T^7 + 3*T^3 + T^2 + 8, 15*T^3 + 15*T^2 + 17*T + 17 ] ####################################################### ####################################################### #### #### 20. BEISPIEL ###################################################### ###################################################### k := GF(29,3); kx := PolynomialAlgebra(k); kT := RationalFunctionField(k); kTy := PolynomialAlgebra(kT); AssignNames_ (kTy, ["y"]); y := kTy.1; AssignNames_ (kT, ["T"]); T := kT.1; f := y^11+11*y^5*T^3+3*y^3*T^7+4*T*y^2+T^11+T^7+12*T^2+7*T+4; F := FunctionField(f); o := MaximalOrderFinite(F); ot := PolynomialAlgebra(o); t := ot.1; AssignNames_ (ot, ["t"]); h1 := t^2+Coerce(o,[3*T^33+1,T^6,12*T^11,0,0,0,0,0,0,0,0]); h2 := t-Coerce(o,[2*T^3+2*T^2+T+2,1,2*T^3+1,12*T+13,0,0,0,0,0,0,0]); h3 := t^2+Coerce(o,[2*T^3+T^2+5,3*T^3+2*T^2+1,T+1,0,0,0,0,0,0,0,0]); h4 := t+Coerce(o,[2,13*T+T^2+1,1,0,0,0,0,0,0,0,0]); h5 := t+Coerce(o,[12,15*T+3*T^2+1,1,0,0,0,0,0,0,0,0]); g := h1*h2*h3*h4*h5;; Coefgr(g,f,F);[7,43]; Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ 27*T^3 + 27*T^2 + 28*T + 27, 28, 27*T^3 + 28, 17*T +16,0, 0, 0,0,0,0,0]],5,1089,1 ] Time: 708.288832 s Globalkurz(k,f,g,F,kx,kT,kTy,o,ot); [ [ t + [ 12, 3*T^2 + 15*T + 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ], 4, 1089, 1 ] Time: 841.07933 s Das Polynom g hat folgende Gestalt: *********************************** t^7 + [ 27*T^3 + 27*T^2 + 28*T + 12, 4*T^2 + 28*T + 1, 27*T^3 + 1, 17*T +16,0,0,0,0,0,0,0]*t^6 + [ 3*T^33 + 3*T^3 + 2*T^2 + 15*T + 2, T^6 + 21*T^5 + 23*T^4 + 26*T^3 + 9*T^2 + 12*T + 26, 12*T^11 + 3*T^4 + 22*T^3 + 17*T^2 + 28*T + 25, 21*T^5 + 2*T^4 + 25*T^3 + 6*T + 19, 6*T^3 + 18*T^2 + 18*T + 2, 5*T + 3, 0, 0, 0, 0, 0 ]*t^5 + [ 23*T^36 + 23*T^35 + 26*T^34 + 7*T^33 + 25*T^6 + 23*T^5 + 25*T^4 + 21*T^3 + 10*T^2 + 28*T + 24, 12*T^35 + 26*T^34 + 3*T^33 + 27*T^9 + 27*T^8 + 28*T^7 + 6*T^6 + 20*T^5 + 22*T^4 + 21*T^3 + 13*T + 24, 23*T^36 + 3*T^33 + 5*T^14 + 5*T^13 + 17*T^12 + 28*T^11 + 4*T^8 + 22*T^7 + 28*T^6 + 23*T^5 + 11*T^4 + 10*T^3 + 23*T^2 + 14*T + 9, 22*T^34 + 19*T^33 + 19*T^13 + 17*T^12 + 12*T^11 + 27*T^9 + 24*T^6 + 10*T^5 + T^4 + 24*T^3 + 25*T^2 + 6*T + 14, 5*T^14 + 12*T^11 + 11*T^7 + 24*T^6 + 8*T^5 + 19*T^4 + 10*T^3 + 14*T + 19, T^12 + 18*T^11 + 14*T^5 + 11*T^4 + 9*T^3 + 19*T^2 + 12*T + 21, 8*T^3 + 18*T^2 + 18*T + 2, 17*T + 16, 0, 0, 0 ]*t^4 + [ 9*T^36 + 6*T^35 + 16*T^34 + 3*T^33 + 2*T^6 + 3*T^5 + 2*T^4 + 15*T^3 + 3*T^2 + 3*T + 10, 5*T^38 + 11*T^37 + 20*T^36 + 27*T^35 + 7*T^34 + 20*T^33 + 3*T^9 + 15*T^8 + 24*T^7 + 15*T^6 + 23*T^5 + 13*T^4 + 19*T^3 + 4*T^2 + 2, 9*T^37 + 8*T^36 + 22*T^35 + 26*T^34 + 17*T^33 + 7*T^14 + 24*T^13 + 6*T^12 + 4*T^11 + 23*T^10 + 26*T^9 + 14*T^8 + 13*T^7 + 14*T^6 + 28*T^5 + 18*T^4 + 12*T^3 + 25*T^2 + 12*T + 21, 5*T^38 + 6*T^37 + 17*T^36 + 18*T^34 + 28*T^33 + 20*T^16 + 15*T^15 + 22*T^14 + 21*T^13 + 28*T^12 + 22*T^11 + 3*T^10 + 22*T^9 + T^8 + 4*T^7 + 25*T^6 + 14*T^4 + 24*T^3 + 16*T^2 + 13*T + 18, 18*T^36 + 25*T^35 + 25*T^34 + 6*T^33 + 7*T^15 + 3*T^14 + T^13 + 17*T^12 + 2*T^11 + 2*T^10 + 25*T^9 + 5*T^8 + 25*T^7 + 23*T^6 + 12*T^4 + 17*T^3 + 18*T^2 + 20*T + 26, 15*T^34 + 9*T^33 + 20*T^16 + 24*T^15 + 10*T^14 + 14*T^12 + 25*T^11 + 6*T^9 + 18*T^8 + 18*T^7 + 12*T^6 + 2*T^5 + 11*T^4 + 26*T^3 + 2*T^2 + 15*T + 10, 14*T^14 + 13*T^13 + 13*T^12 + 24*T^11 + 5*T^7 + 3*T^6 + 21*T^4 + 14*T^3 + 13*T^2 + 25*T + 5, 2*T^12 + 7*T^11 + 5*T^2 + 8*T + 3, 0, 0, 0 ]*t^3 + [ 17*T^39 + 11*T^38 + 17*T^37 + 11*T^36 + 7*T^35 + 7*T^33 + 16*T^6 + 24*T^5 + 16*T^4 + 11*T^3 + 14*T^2 + 25*T + 4, 11*T^39 + 2*T^38 + 8*T^37 + 5*T^36 + 17*T^35 + 13*T^34 + 11*T^33 + 25*T^12 + 23*T^11 + 25*T^10 + 23*T^9 + 27*T^8 + 18*T^7 + 10*T^6 + 16*T^5 + 11*T^4 + 14*T^3 + 3*T + 5, 11*T^40 + 23*T^39 + 11*T^38 + 4*T^37 + 7*T^36 + 11*T^35 + 13*T^34 + 24*T^33 + 10*T^17 + 15*T^16 + 10*T^15 + 15*T^14 + 28*T^13 + 23*T^12 + 19*T^11 + 10*T^10 + 19*T^9 + 3*T^8 + 16*T^7 + 16*T^6 + 6*T^5 + 23*T^4 + 3*T^3 + T^2 + 11*T + 3, 11*T^39 + T^38 + 3*T^37 + 14*T^36 + 17*T^35 + 25*T^34 + 23*T^33 + 15*T^17 + 8*T^16 + 3*T^15 + 20*T^14 + 4*T^13 + 21*T^12 + 9*T^11 + 22*T^10 + 6*T^9 + 7*T^8 + 6*T^7 + 18*T^6 + 20*T^5 + 11*T^4 + 16*T^3 + 20*T^2 + 3*T + 9, 11*T^40 + 24*T^39 + 24*T^38 + 28*T^37 + T^36 + 13*T^34 + 28*T^33 + 15*T^18 + 5*T^17 + 15*T^16 + 16*T^15 + 28*T^14 + 15*T^13 + 17*T^12 + 19*T^11 + 18*T^10 + 5*T^9 + 27*T^8 + 25*T^6 + 17*T^5 + 11*T^4 + 11*T^3 + 20*T^2 + 8*T + 22, 13*T^38 + 4*T^37 + 27*T^36 + 28*T^35 + 7*T^34 + 5*T^33 + 15*T^17 + 4*T^16 + 12*T^15 + 27*T^14 + 4*T^13 + 21*T^12 + 13*T^11 + T^10 + 22*T^9 + 10*T^8 + 28*T^7 + 3*T^6 + 24*T^5 + 2*T^4 + 3*T^3 + T^2 + 13*T + 22, 24*T^36 + 25*T^35 + 25*T^34 + 6*T^33 + 15*T^18 + 9*T^17 + 9*T^16 + 25*T^15 + 4*T^14 + 23*T^12 + 10*T^11 + 11*T^10 + 9*T^9 + 26*T^8 + 17*T^7 + 15*T^6 + 2*T^5 + 22*T^4 + 8*T^3 + 28*T^2 + 27*T + 19, 22*T^34 + 19*T^33 + 23*T^16 + 16*T^15 + 21*T^14 + 25*T^13 + 28*T^12 + 20*T^11 + 8*T^9 + 18*T^8 + 18*T^7 + 11*T^6 + 8*T^5 + 28*T^4 + 23*T^3 + 19*T^2 + 17*T + 16, 9*T^14 + 13*T^13 + 13*T^12 + 24*T^11 + 17*T^7 + 16*T^6 + T^4 + 21*T^3 + 10*T^2 + 8*T + 18, T^12 + 18*T^11 + 17*T^2 + 4*T + 16, 0 ]*t^2 + [ 6*T^39 + 9*T^38 + 6*T^37 + 7*T^36 + 3*T^35 + 22*T^34 + 27*T^33 + 2*T^6 + 3*T^5 + 2*T^4 + 12*T^3 + T^2 + 17*T + 9, 10*T^41 + 27*T^40 + 13*T^39+ 6*T^38 + 28*T^37 + 8*T^36 + 14*T^35 + 22*T^34 + 15*T^33 + 2*T^12 + 3*T^11 + 2*T^10 + 12*T^9 + 14*T^8 + 26*T^7 + 23*T^6 + 2*T^5 + 19*T^4 + 22*T^3 + 24*T^2 + 17*T + 5, 15*T^41 + 3*T^40 + 22*T^39 + 26*T^38 + 16*T^37 + 28*T^36 + 24*T^35 + 10*T^34 + 17*T^33 + 24*T^17 + 7*T^16 + 24*T^15 + 12*T^14 + 21*T^13 + 15*T^12 + 23*T^11 + 19*T^10 + 22*T^9 + 18*T^7 + 22*T^6 + 28*T^5 + 15*T^4 + 19*T^3 + 8*T^2 + 13*T + 25, 10*T^41 + 15*T^40 + 24*T^38 + 7*T^37 + 26*T^36 + 19*T^35 + 21*T^34 + 26*T^33 + 11*T^19 + 21*T^18 + 23*T^17 + 24*T^16 + 25*T^15 + 8*T^14 + 28*T^13 + 18*T^12 + T^11 + 15*T^10 + 19*T^9 + 21*T^8 + 18*T^7 + 25*T^6 + 8*T^5 + 12*T^4 + 28*T^3 + 16*T^2 + 7*T + 28, 15*T^41 + 28*T^40 + 12*T^39 + 7*T^37 + 4*T^36 + 6*T^34 + 14*T^33 + 2*T^19 + 12*T^18 + T^17 + 17*T^16 + 6*T^15 + 9*T^14 + 14*T^13 + 11*T^12 + 18*T^11 + 12*T^10 + 28*T^9 + 21*T^8 + 26*T^7 + 3*T^6 + 12*T^4 + 11*T^3 + 2*T + 24, T^39 + 6*T^38 + 4*T^37 + 20*T^36 + 6*T^35 + T^34 + 21*T^33 + 11*T^19 + 2*T^18 + 9*T^16 + 28*T^15 + 22*T^14 + 8*T^13 + T^12 + 17*T^11 + 12*T^10 + 11*T^9 + 2*T^7 + 5*T^6 + 2*T^5 + 11*T^4 + 26*T^3 + 2*T^2 + 10*T + 7, 5*T^37 + 13*T^36 + 10*T^35 + 17*T^34 + 15*T^33 + 2*T^19 + 25*T^18 + 19*T^17 + 28*T^15 + 16*T^14 + 5*T^12 + 11*T^10 + 26*T^9 + 2*T^8 + 10*T^7 + 7*T^6 + 21*T^4 + 14*T^3 + 13*T^2 + 25*T + 5, 15*T^35 + 24*T^34 + 9*T^33 + 4*T^17 + 24*T^16 + 16*T^15 + 22*T^14 + 24*T^13 + 4*T^12 + 26*T^11 + 21*T^10 + 14*T^9 + 13*T^8 + 25*T^7 + 5*T^6 + 5*T^2 + 8*T + 3, 20*T^15 + 23*T^14 + 11*T^13 + 10*T^12 + 2*T^11 + 5*T^8 + 8*T^7 + 3*T^6,2*T^13+9*T^12+7*T^11,0]*t + [ 2*T^39 + 3*T^38 + 2*T^37 + 22*T^36 + 6*T^35 + 17*T^34 + 5*T^33 + 28*T^24 + 10*T^23 + 11*T^22 + 28*T^20 + 10*T^19 + 11*T^18 + 17*T^15 + 26*T^14 + 24*T^13 + T^12 + 15*T^11 + 20*T^6 + T^5 + 20*T^4 + 17*T^3 + 2*T^2 + 25*T + 21, 16*T^41 + 25*T^40 + 12*T^39 + 17*T^38 + 25*T^37 + 8*T^36 + 12*T^35 + 25*T^34 + 4*T^33 + 20*T^12 + T^11 + 20*T^10 + 17*T^9 + 17*T^8 + 14*T^7 + 25*T^6 + 25*T^5 + 18*T^4 + 22*T^3 + 4*T^2 + 18*T + 11, 22*T^43 + 23*T^42 + 21*T^41 + 24*T^40 + 14*T^39 + 7*T^38 + 7*T^37 + 2*T^36 + 21*T^35 + 20*T^34 + 14*T^33 + 8*T^17 + 12*T^16 + 8*T^15 + 12*T^14 + 24*T^13 + 16*T^11 + 6*T^10 + 20*T^9 + 11*T^8 + 26*T^7 + 6*T^6 + 12*T^5 + 12*T^4 + 20*T^3 + 7*T^2 + 26*T + 24, 4*T^43 + 11*T^42 + 10*T^41 + 5*T^40 + 19*T^39 + T^38 + T^37 + 5*T^36 + 14*T^35 + 13*T^34 + 4*T^33 + 26*T^20 + 7*T^19 + 17*T^18 + 19*T^17 + 27*T^16 + 11*T^15 + 10*T^14 + 27*T^13 + 8*T^12 + 28*T^11 + 23*T^10 + 14*T^9 + 20*T^8 + 18*T^7 + 11*T^6 + 10*T^5 + 10*T^4 + 21*T^3 + 24*T^2 + 14*T + 11, 22*T^43 + T^42 + 6*T^41 + 22*T^40 + 28*T^39 + 27*T^38 + 5*T^37 + 3*T^36 + 2*T^35 + 11*T^34 + 9*T^33 + T^21 + 5*T^20 + 26*T^19 + 9*T^18 + 27*T^17 + 10*T^16 + 22*T^15 + 21*T^14 + 18*T^13 + 9*T^12 + 8*T^11 + 27*T^10 + 2*T^9 + 26*T^8 + 2*T^7 + T^6 + 9*T^5 + 21*T^4 + T^3 + 20*T^2 + 23*T + 3, 4*T^43 + 7*T^42 + T^41 + 13*T^40 + 10*T^39 + T^38 + 2*T^37 + 11*T^36 + 4*T^35 + 3*T^34 + 3*T^33 + 16*T^21 + 15*T^20 + 11*T^19 + 20*T^18 + 18*T^17 + 10*T^16 + 8*T^15 + 27*T^14 + 15*T^13 + 13*T^12 + 25*T^11 + 3*T^10 + 13*T^9 + T^8 + 8*T^7 + 16*T^6 + 10*T^5 + 20*T^4 + 23*T^3 + 11*T^2 + T + 1, 21*T^41 + 15*T^40 + 11*T^39 + 6*T^38 + 8*T^37 + T^35 + 27*T^34 + 22*T^33 + T^21 + 4*T^20 + 24*T^19 + T^18 + 25*T^17 + 3*T^16 + 3*T^15 + 22*T^14 + 22*T^13 + 28*T^12 + 17*T^11 + 20*T^10 + 23*T^9 + 18*T^8 + 6*T^7 + 24*T^6 + 2*T^5 + 22*T^4 + 10*T^2 + 9*T + 17, 27*T^39 + 24*T^38 + 26*T^37 + 11*T^36 + 28*T^35 + 16*T^21 + 28*T^20 + 4*T^19 + 23*T^18 + 11*T^17 + 4*T^16 + 8*T^15 + 22*T^14 + 21*T^13 + 6*T^12 + 14*T^11 + 22*T^10 + 10*T^8 + 9*T^7 + 26*T^6 + 8*T^5 + 28*T^4 + 23*T^3 + 19*T^2, 3*T^37 + 5*T^36 + T^35 + 24*T^34 + 25*T^33 + 26*T^19 + 2*T^18 + 15*T^17 + 24*T^16 + 3*T^15 + 4*T^13 + T^12 + 9*T^11 + 28*T^10 + 23*T^9 + 19*T^8 + T^4 + 21*T^3 + 10*T^2 + 8*T + 18, 22*T^35 + 12*T^34 + 19*T^33 + 21*T^17 + 9*T^16 + 17*T^15 + 15*T^14 + 25*T^13 + T^10 + 21*T^9 + 10*T^8 + 8*T^7 + 18*T^6 + 17*T^2 + 4*T + 16, 12*T^15 + 20*T^14 + 4*T^13 + 9*T^12 + 13*T^11 + 17*T^8 + 4*T^7 + 16*T^6 ]