The KASH Reference Manual

(still) with TeX-inline notation

AbelianDualGroup

Returns the dual group of a given finite abelian group.

Syntax:

d := AbelianDualGroup(g);

groups

  g, d

AbelianDualHom

Creates the dual homomorphism of a given homomorphism between two finite Abelian groups.

Syntax:

dualhom := AbelianDualHom(hom);

homomorphism

hom

homomorphism from g1 to g2

AbelianFieldToRCF

Computes the data neccessary to get an abelian field as a Ray Class Field.

Syntax:

rc := AbelianFieldToRCF(o [, I]);

list	rc	[ ideal, inf, relations ]
order	o	of an abelian field with known automorphisms
ideal	I	of the coef. ring of `o`. Must be a multiple of the conductor.

AbelianFixPointGroup

Computes the fix point group of endomorphisms.

Syntax:

f := AbelianFixPointGroup(hom);
f := AbelianFixPointGroup(L);

group	f
homomorphism	hom
list	L	list of homomorphisms

AbelianGroupBasis

Returns a basis of a finite abelian group.

Syntax:

L := AbelianGroupBasis(g [, exp]);

group	g
list	L	list of basis elements of g
boolean	exp

AbelianGroupCanonicalQuotient

Returns the canonical quotient group of a homomorphism.

Syntax:

q := AbelianGroupCanonicalQuotient(hom);

group	q
homomorphism	hom

AbelianGroupCreate

Creates a finite abelian group via a given relation matrix.

Syntax:

g := AbelianGroupCreate(mat);
g := AbelianGroupCreate(mat, "*");
g := AbelianGroupCreate(mat, "+");

group	g
matrix or list of lists	mat

AbelianGroupCyclicFactors

Returns cyclic factors and their orders from the Smith normal form.

Syntax:

L := AbelianGroupCyclicFactors(g);

group	g
list	L	list of cyclic factors of g

AbelianGroupDirectProduct

Returns the direct product of two finite abelian groups.

Syntax:

d := AbelianGroupDirectProduct(g1, g2);
d := AbelianGroupDirectProduct(L);

groups	g1, g2, d
list	L	list of groups

AbelianGroupDiscreteExp

Returns the object corresponding to a representation in a group.

Syntax:

b := AbelianGroupDiscreteExp(a);

group	g
group element	a	representation of b in the abstract group g
object or boolean	b

AbelianGroupDiscreteLog

Returns the representation of an object in a group.

Syntax:

a := AbelianGroupDiscreteLog(g, b);

group	g
group element	a	representation of b in the abstract group g
object	b

AbelianGroupEltCreate

Creates an element of a group.

Syntax:

elt := AbelianGroupEltCreate(g, vector);

group element	elt
group	g
list or matrix with one row of integers	vector

AbelianGroupEltMove

Moves an element from a subgroup into a supergroup.

Syntax:

elt2 := AbelianGroupEltMove(elt1, g);

group	g
group element	elt1
group element	elt2	element of the group g

AbelianGroupEltOrder

Returns the order of an element.

Syntax:

a := AbelianGroupEltOrder(elt);

group element	elt
integer	a

AbelianGroupEltRandom

Generates a random element of an abelian group.

Syntax:

e := AbelianGroupEltRandom(G);

AbelianGroupElt	e
AbelianGroup	G

AbelianGroupEltReduce

Returns a reduced representation of a group element.

Syntax:

elt2 := AbelianGroupEltReduce(elt1 [, positive]);

group element	elt1
group element	elt2	reduced representation
boolean	positive

AbelianGroupEnumInit

Generates a environment for the enumeration of all elements of a finite abelian group G.

Syntax:

s := AbelianGroupEnumInit(G [, "gen"]);

record	s
AbelianGroup	G
arbitray	gen	if present, only a set of generators will be enumerated.

AbelianGroupEnumNext

Computes "the next" element of an abelian group.

Syntax:

flag := AbelianGroupEnumNext(s);

boolean	flag	`true` iff there is a valid element in `s`
record	s	generated by `AbelianGroupEnumInit`
AbelianGroupElt	s.elt

AbelianGroupEqual

Tests if two groups are equal.

Syntax:

test := AbelianGroupEqual(g1, g2);

groups	g1, g2
boolean	test

AbelianGroupExponent

Returns the exponent of a group.

Syntax:

a := AbelianGroupExponent(g);

group	g
integer	a

AbelianGroupGenerators

Returns generators of a group.

Syntax:

L := AbelianGroupGenerators(g [, exp]);

group	g
list	L	list of generators of g
boolean	exp

AbelianGroupHomCreate

Creates the homomorphism corresponding to a matrix.

Syntax:

hom := AbelianGroupHomCreate(g1, g2, mat [, check]);
hom := AbelianGroupHomCreate(g1, g2, mat, [matinv]);

groups	g1, g2
homomorphism	hom	homomorphism from g1 to g2
boolean	check
matrix	matinv

AbelianGroupHomImage

Returns the image of a group homomorphism.

Syntax:

g := AbelianGroupHomImage(hom [, generators]);

group	g
homomorphism	hom
boolean	generators

AbelianGroupHomKernel

Returns the kernel of a group homomorphism.

Syntax:

g := AbelianGroupHomKernel(hom [,generators]);

group	g
homomorphism	hom
boolean	generators

AbelianGroupIndex

Returns the index of a subgroup in a group.

Syntax:

a := AbelianGroupIndex(g, s);

group	g
group	s	subgroup of g
integer	a

AbelianGroupIntersect

Returns the intersection of two groups.

Syntax:

g := AbelianGroupIntersect(g1, g2);

groups

  g, g1, g2

AbelianGroupIsAut

Tests whether a function is a group automorphism.

Syntax:

L := AbelianGroupIsAut(hom [, check]);

homomorphism	hom
boolean	check
list	L

AbelianGroupIsSub

Tests whether s is a subgroup of g.

Syntax:

test := AbelianGroupIsSub(s, g);

groups	g, s
boolean	test

AbelianGroupMinNumberGenerators

Returns the minimal number of generators of a group.

Syntax:

a := AbelianGroupMinNumberGenerators(g);

group	g
integer	a

AbelianGroupMultiHomCreate

Creates a multilinear mapping.

Syntax:

f := AbelianGroupMultiHomCreate(d, g, t, mat);

group	d, g, t
matrix or list of lists	mat

AbelianGroupName

Gives a name to a group.

Syntax:

AbelianGroupName(g, s);

group	g
string	s	name of g

AbelianGroupNumberGenerators

Returns the number of generators.

Syntax:

a := AbelianGroupNumberGenerators(g);

group	g
integer	a

AbelianGroupOrder

Returns the order of a group.

Syntax:

a := AbelianGroupOrder(g);

group	g
integer	a

AbelianGroupSmithCreate

Returns the smith normal form of a group.

Syntax:

s := AbelianGroupSmithCreate(g [, generators]);

group	g, s
boolean	generators

AbelianGroupTensorProduct

Returns the tensor product of groups.

Syntax:

t := AbelianGroupTensorProduct(L);

group	t
list	L	list of groups

AbelianGroupUnite

Returns the smallest group containing two given groups.

Syntax:

g := AbelianGroupUnite(g1, g2);

groups

  g, g1, g2

AbelianHomGroup

Returns the group consisting of homomorphisms between two groups.

Syntax:

h := AbelianHomGroup(g1, g2);

groups

  g1, g2, h

AbelianMultiHomGroup

Computes the group of multilinear mappings.

Syntax:

L := AbelianMultiHomGroup([g_{1}, … ,g_{n}], h);

groups	g_{1}, … , g_{n}, h
list	L

AbelianQuotientGroup

Returns the quotient of a group and a subgroup.

Syntax:

q := AbelianQuotientGroup(g, s);

groups	g, q
group	s	subgroup of g

AbelianRayClassGroupAutoCreate

Given a field automorphism, this function generates the corresponding automorphism of abelian groups.

Syntax:

aut := AbelianRayClassGroupAutoCreate(G, sigma);

AbelianGroupHom	aut
AbelianGroup	G	must be RayClassGroupToAbelianGroup
KASH function	sigma	acting on ideals

AbelianRayGroupImbed

Imbeds a ClassGroup into a RayClassGroup.

Syntax:

dualhom := AbelianRayGroupImbed(G,g);

abstract rayclassgroup	G
abstract classgroup	g

AbelianSubGroup

Creates a subgroup.

Syntax:

s := AbelianSubGroup([g,] L [, generators]);
s := AbelianSubGroup(g, mat [, generators]);

groups	g, s
list	L	list of elements of g or list of exponent vectors
matrix	mat
boolean	generators

Abs

Returns the absolute value of a number.

Syntax:

a := Abs(x);

complex	a
complex	x

Alff

Creates an algebraic function field F/k(T).

Syntax:

F := Alff(f);

algebraic function field	F
polynomial	f

AlffCanonicalDivisor

Computes a canonical divisor of an algebraic function field F/k.

Syntax:

W := AlffCanonicalDivisor(F);

alff divisor	W
algebraic function field	F

AlffClassGroup

Computes the structure of the divisor class group of a global function field.

Syntax:

Cl := AlffClassGroup( F [, b ] [, A ] [, "fast" ] );

list	Cl	of integer and list (class group structure)
alff	F	global function field
integer	b	bound for degree of places to be used
alff divisor	A	for reduction

AlffClassGroupGenBound

Returns a bound for the degree of prime divisors which generate the divisor class group.

Syntax:

b := AlffClassGroupGenBound(q, g);
b := AlffClassGroupGenBound(F);

integer	b	generation bound
integers	q, g	exact constant field size and genus of F
alff	F	global function field

AlffClassGroupGenBoundStrong

Return a bound for the degree of prime divisors which generate the divisor class group.

Syntax:

b := AlffClassGroupGenBoundStrong(F);

integer	b	generation bound
alff	F	global function field

AlffClassGroupGens

Computes generators of the divisor class group of a global function field.

Syntax:

L := AlffClassGroupGens( F, [ b, ] [ A, ] [ "fast" ] );

list	L	of generating divisors
alff	F	global function field
integer	b	bound for degree of places to be used
alff divisor	A	for reduction

AlffClassGroupPRank

Computes the p-rank of the class group of a global function field.

Syntax:

s := AlffClassGroupPRank(F);

integer	s
global function field	F

AlffClassNumberApprox

Compute an approximation of the class number of a global function field.

Syntax:

hbar := AlffClassNumberApprox(F, b);

real	hbar	approximated class number
alff	F	global function field
integer	b	bound

AlffClassNumberApproxBound

Return a bound for the prime divisors to be considered for the approximation of the class number of a global function field.

Syntax:

b := AlffClassNumberApproxBound(q, g, a);
b := AlffClassNumberApproxBound(F, a);

integer	b	approximation bound
integers	q, g	exact constant field size and genus
alff	F	global function field
real	a	\in R^{> 1}

AlffConstField

Returns the constant field k of definition of an algebraic function field F/k(T).

Syntax:

k := AlffConstField(F);

field	k
algebraic function field	F

AlffConstFieldSize

Returns the size of the constant field of definition of a global function field.

Syntax:

q := AlffConstFieldSize(F);

integer	q	size of constant field
algebraic function field	F

AlffDeg

Returns the degree of the extension F/k(T).

Syntax:

n := AlffDeg(F);

integer	n
algebraic function field	F

AlffDiff

Returns the exact differential dh of an alff element h.

Syntax:

dh := AlffDiff(h);

alff differential	dh
alff element	h

AlffDiffDivisor

Computes the divisor of an alff differential hdT.

Syntax:

D := AlffDiffDivisor(hdT);

alff divisor	D
alff differential	hdT

AlffDiffResiduum

Computes the residuum of a differential.

Syntax:

r := AlffDiffResiduum(P, hdT);

field element	r	the residuum of hdT at P
alff place	P
alff differential	hdT

AlffDiffSpace

Computes a basis for the space of Weil differentials Omega(D) := {\; omega \in Omega(F/k) \;|\; (omega) >= D \;} for a divisor D of an algebraic function field F/k.

Syntax:

B := AlffDiffSpace(D);

list	B	of alff differentials
alff divisor	D

AlffDiffValuation

Returns the order of a non zero alff differential at a place.

Syntax:

v := AlffDiffValuation(p, fdx);

integer	v
alff place	p
alff differential	fdx

AlffDifferent

Computes the different of an algebraic function field extension F/ k(T).

Syntax:

D := AlffDifferent(F);

alff divisor	D
algebraic function field	F

AlffDifferentDeg

Computes the degree of the different of an algebraic function field extension F/ k(T).

Syntax:

d := AlffDifferentDeg(F);

integer	d
algebraic function field	F

AlffDifferentiation

Computes higher differentiations of function field elements.

Syntax:

b := AlffDifferentiation(p, m, a);

alff element	b	m-times differentiated element
integer	m
alff place or element	p
alff element	a	element to differentiate

AlffDimExactConstField

Returns the k-dimension of the exact constant field \widetilde k of an algebraic function field F/k(T).

Syntax:

l := AlffDimExactConstField(F);

integer	l
algebraic function field	F

AlffDivisor

Creation of divisors.

Syntax:

D := AlffDivisor(F);
D := AlffDivisor(a);
D := AlffDivisor(P);
D := AlffDivisor(I, J);
D := AlffDivisor(u, v);

alff divisor	D
algebraic function field	F
alff order element	a
alff place	P
alff order ideal	I	of the finite maximal order
alff order ideal	J	of the infinite maximal order
alff order elements	u,v

AlffDivisorAlff

Given a divisor this function returns the algebraic function field it belongs to.

Syntax:

F := AlffDivisorAlff(D);

algebraic function field	F
alff divisor	D

AlffDivisorClassRep

Find the class representation of a divisor class in given generators.

Syntax:

L := AlffDivisorClassRep(D);

list	L	exponents
alff divisor	D

AlffDivisorDeg

Degree of a divisor.

Syntax:

d := AlffDivisorDeg(D);

integer	d
alff divisor	D

AlffDivisorDegOne

Computes a divisor of degree one for a global function field.

Syntax:

D := AlffDivisorDegOne(F);

alff divisor	D	of degree one
global function field	F

AlffDivisorDen

Returns the denominator of an alff divisor.

Syntax:

D2 := AlffDivisorDen(D);

alff divisor	D2
alff divisor	D

AlffDivisorIdeals

Divisor corresponding ideals of maximal orders.

Syntax:

L := AlffDivisorIdeals(D);

list	L	of finite and infinite ideals
alff divisor	D

AlffDivisorLBasis

Computes a basis of a Riemann-Roch space

Syntax:

B := AlffDivisorLBasis(D);

list	B	of basis elements
alff divisor	D

AlffDivisorLBasis

Computes a basis of a Riemann-Roch space

Syntax:

B := AlffDivisorLBasis(D);

list	B	of basis elements
alff divisor	D

AlffDivisorLBasisShort

Computes a basis of a Riemann-Roch space.

Syntax:

B := AlffDivisorLBasisShort(D);

list	B	of pairs of basis elements b_i and degree bounds d_i
alff divisor	D

AlffDivisorLDim

Dimension of a Riemann-Roch space.

Syntax:

l := AlffDivisorLDim(D);

integer	l
alff divisor	D

AlffDivisorLIndex

Index of speciality of alff divisors.

Syntax:

i := AlffDivisorLIndex(D);

integer	i
divisor	D

AlffDivisorLargeLBasisShort

Computes a basis of a large divisor.

Syntax:

B := AlffDivisorLargeLBasis(D, "raw");

list	B	the short basis
alff divisor	D

AlffDivisorLargeLDim

Computes the dimension of a large divisor.

Syntax:

d := AlffDivisorLargeLDim(D);

integer	d	the dimension
alff divisor	D

AlffDivisorNorm

Computes the norm of a divisor.

Syntax:

N := AlffDivisorNorm(D);

qf element	N
alff divisor	D

AlffDivisorNum

Returns the numerator of an alff divisor.

Syntax:

D1 := AlffDivisorNum(D);

alff divisor	D1
alff divisor	D

AlffDivisorPlaces

Places and exponents of an alff divisor.

Syntax:

L := AlffDivisorPlaces(D);

list	L	of lists of alff places and integers
alff divisor	D

AlffDivisorReduction

Computes a reduced divisor.

Syntax:

L := AlffDivisorReduction(D);
L := AlffDivisorReduction(D, A);

list	L	as above
alff divisor	D	to reduce
alff divisor	D

AlffDivisorSupp

Returns the support of a divisor as a list of places.

Syntax:

L := AlffDivisorSupp(D);

list	L	of places
alff divisor	D

AlffDivisorsSmoothNum

Compute the number of (n, m)-smooth divisors.

Syntax:

N := AlffDivisorsSmoothNum(n, m, P);

integer	N	number of (n, m)-smooth divisors
integers	n, m
list	P	of integers

See also: AlffInit, AlffOrders, AlffPlacesNum

AlffDivisorsSmoothRatio

Return a ratio of numbers of smooth divisors times a number of places.

Syntax:

r := AlffDivisorsSmoothRatio(n, m, P);

real	r	smoothness ratio
integers	n, m
list	P	of integers

AlffEllipticFunField

Creates an elliptic algebraic function field F/k.

Syntax:

F := AlffEllipticFunField(k, L);

algebraic function field	F
field	k
list	L

AlffElt

Creates an element of an algebraic function field F/k(T).

Syntax:

a := AlffElt(o, s);
a := AlffElt(o, b);
a := AlffElt(o, L);

algebraic function field element	a
algebraic function field order	o
finite field element, rational or order element	s	a constant
polynomial or quotient field element	b	a rational function
list	L	of above s or b of length n

AlffEltAlff

Returns the algebraic function field of an alff element.

Syntax:

F := AlffEltAlff(a);

alff	F
alff element	a

AlffEltApprox

Computes an approximating element of an algebraic function field F.

Syntax:

alpha := AlffEltApprox(S, Lambda, A [,a]);

element	alpha	of the algebraic function field F
list	S	of places of F
alff divisor	A
list	a	of elements of F

AlffEltBstar

Computes B^*(a) for a global function field element a.

Syntax:

q := AlffEltBstar(a);

rational	q
global function field element	a

AlffEltCharPoly

Returns the characteric polynomial of an algebraic function field element.

Syntax:

p := AlffEltMinPoly(a);

polynomial	p
alff element	a

AlffEltDen

Returns the denominator of an algebraic function field element.

Syntax:

d := AlffEltDen(a);

quotient field element or polynomial	d
algebraic function field element	a

See also: AlffEltNum, AlffEltToList, Num, Den

AlffEltEval

Evaluates an algebraic function at a place.

Syntax:

b := AlffEltEval(P, a);

field element	b	value of a at P
alff place	P
alff element	a

AlffEltGenT

Returns the indeterminate T as maximal order element.

Syntax:

T := AlffEltGenT(F);

algebraic function field	F
alff element	T

AlffEltGenY

Returns the indeterminate y as finite maximal order element.

Syntax:

Y := AlffEltGenY(F);

algebraic function field	F
alff element	Y

AlffEltInftyVals

Computes the inequivalent valuations of a global function field element at infinity.

Syntax:

L := AlffEltInftyVals(a);

list	L
global function field element	a

See also: AlffPlaceSplit, AlffSignature, Alff

AlffEltIsInIdeal

Checks whether a function field element is in an ideal.

Syntax:

b := AlffEltIsInIdeal(a, I);

boolean	b
alff order element	a
alff order ideal	I

AlffEltMin

Returns the denominator d of an alff element a from a form of the representation matrix of da \in o.

Syntax:

m := AlffEltMin(a);

coefficient ring element	m
alff element	a

AlffEltMinPoly

Returns the minimal polynomial of an algebraic function field element.

Syntax:

p := AlffEltMinPoly(a);

polynomial	p
alff element	a

AlffEltMove

Moves an algebraic function field element between orders.

Syntax:

a := AlffEltMove(b, o);

algebraic function field elements	a,b
algebraic function field order	o

AlffEltNewtonLift

Lifts an algebraic element with the Newton lifting method.

Syntax:

beta:=AlffEltNewtonLift(o, g, alpha, p, k, den);

algebraic function field element	beta
algebraic function field order	o
polynomial	g
algebraic function field element	alpha
polynomial	p
integer	k
polynomial	den

AlffEltNorm

Computes the norm of an element of an algebraic function field.

Syntax:

a := AlffEltNorm(b);

quotient field element or polynomial	a
algebraic function field element	b

See also: AlffTrace, AlffEltCharPoly, Norm, Trace

AlffEltNum

Returns the numerator of an algebraic function field element.

Syntax:

b := AlffEltNum(a);

algebraic function field element	a
algebraic function field element	b

See also: AlffEltDen, AlffEltToList, Num, Den

AlffEltOrder

Returns the order with respect to which the element has been defined.

Syntax:

o := AlffEltOrder(a);

algebraic function field order	o
algebraic function field element	a

See also: AlffElt, AlffEltMove, AlffOrderAlff

AlffEltPthRoot

Computes the p^r th root of an element of a global function field.

Syntax:

b := AlffEltPthRoot(a, r);

alff element	b
global function field	F
positive integer	r

AlffEltRepMat

Returns a representation matrix of an algebraic function field element.

Syntax:

M := AlffEltRepMat(a);

matrix	M
alff element	a

AlffEltResiduum

Computes the residuum of an algebraic function.

Syntax:

r := AlffEltResiduum(P, t, a);

field element	r	the residuum of a at P for t
alff place	P
alff element	t	a local paramter at P
alff element	a	an algebraic function

AlffEltRoot

Computes an n-th root of a function field element a.

Syntax:

b := AlffEltRoot(a,n );

alff element	b
alff element	a
integer	n

AlffEltToList

Returns the coefficients and the denominator of an algebraic function field element.

Syntax:

L := AlffEltToList(a);

list	L
algebraic function field element	a

See also: AlffEltNum, AlffEltDen, AlffElt, Num, Den

AlffEltToResField

Returns a representative of the class of an algebraic function in the residue class field of a place.

Syntax:

b := AlffEltToResField(a, P);

field element	b	value of a at P
alff element	a
alff place	P

AlffEltTrace

Computes the trace of an element of an algebraic function field.

Syntax:

a := AlffEltTrace(b);

quotient field element or polynomial	a
algebraic function field element	b

AlffEltValuation

The valuation of an algebraic function field element at a place.

Syntax:

v := AlffEltValuation(P, a);

integer	v
alff place	P
alff order element	a

AlffGapNumbers

Returns the gap numbers of a divisor.

Syntax:

L := AlffGapNumbers( D [, P] );
L := AlffGapNumbers( F [, P] );

list	L	containing the gap numbers
alff divisor	D
alff	F	equivalent to taking D = 0

AlffGenus

missing shortdoc

Syntax:

g := AlffGenus(F);

algebraic function field	F
integer	g	the genus

AlffHasseWittInvariant

Computes the Hasse-Witt invariant of a global function field.

Syntax:

s := AlffHasseWittInvariant(F);

integer	s
global function field	F

AlffHermitianFunField

Creates a Hermitian global function field F/FF_{q^2}(T).

Syntax:

F := AlffHermitianFunField(p, d);

global function field	F
integer	p, d

AlffHermitianFunField

Creates a Hermitian global function field F/FF_{q^2}(T).

Syntax:

F := AlffHermitianFunField(p, d);

global function field	F
integer	p, d

AlffIdealAlff

Returns the algebraic function field of an alff order ideal.

Syntax:

F := AlffIdealAlff(I);

alff	F
alff order ideal	I

AlffIdealBasis

Returns a list containing the basis elements of an alff order ideal.

Syntax:

B := AlffIdealBasis(I);

list	B	of basis elements
alff order ideal	I

AlffIdealBasisUpperHNF

Return an ideal basis in upper Hermite normal form.

Syntax:

L := AlffIdealBasisUpperHNF(I);

list	L
alff order ideal	I

AlffIdealClassGroupUnitsInfty

Compute the unit group and the ideal class group of the finite maximal order {rm Cl}( k[T], F ).

Syntax:

L := AlffIdealClassGroupUnitsInfty(F);

list	L	units and ideal class group info
alff	F	global function field

AlffIdealFactor

Return the factorization of an ideal.

Syntax:

L := AlffIdealFactor(I);

list	L
alff order ideal	I	in maximal order

AlffIdealGenerators

Returns a list of generators of an ideal.

Syntax:

L := AlffIdealGenerators(I);

ideal	I
list	L	of two algebraic numbers

AlffIdealIsPrime

missing shortdoc

Syntax:

b := AlffIdealIsPrime(I);

boolean	b
alff order ideal	i	in maximal order

AlffIdealIsPrimeKnown

missing shortdoc

Syntax:

b := AlffIdealIsPrimeKnown(I);

boolean	b
alff order ideal	I	in maximal order

AlffIdealNorm

missing shortdoc

Syntax:

a := AlffIdealNorm(I);

rational function or polynomial	a
alff order ideal	I

AlffIdealOrder

missing shortdoc

Syntax:

o := AlffIdealOrder(a);

alff order	o
alff order ideal	a

AlffIdealPlace

Given a prime ideal this function returns the corresponding place.

Syntax:

P := AlffIdealPlace(I);

prime ideal	I
alff place	P

AlffIdealValuation

Return the valuation of an ideal at a prime ideal.

Syntax:

v := AlffIdealValuation(P, I);

integer	v
alff order ideals	P, I	in maximal order o

AlffIharaBound

missing shortdoc

Syntax:

b := AlffIharaBound(F);
b := AlffIharaBound(q, g);

global function field	F
integer	q, g
integer	b

AlffInit

Initializes some useful variables for an algebraic function field session.

Syntax:

AlffInit(k);
AlffInit(k, T);
AlffInit(k, T, y);

field	k
strings	T, y	the variable names

AlffInit

Initializes some useful variables for an algebraic function field session.

Syntax:

AlffInit(k);
AlffInit(k, T);
AlffInit(k, T, y);

field	k
strings	T, y	the variable names

AlffIsAbs

Returns whether a function field is an absolute extension.

Syntax:

b := AlffIsAbs(F);

boolean	b
function field	F

AlffIsGlobal

Returns whether an algebraic function field F is global.

Syntax:

b := AlffIsGlobal(F);

boolean	b	whether global or not
algebraic function field	F

AlffIsGlobalAssert

Signals an error if an algebraic function field F is not global.

Syntax:

AlffIsGlobalAssert(F);

algebraic function field

AlffLPoly

Computes the L-polynomial of a global function field.

Syntax:

L := AlffLPoly(F);

polynomial	L
global function field	F

AlffLPolyLift

Lifts the L-polynomial of a global function field to constant field extensions.

Syntax:

Lr := AlffLPolyLift(L, r);

polynomial	Lr, L
integer	r

AlffLPolyRed

Computes the L-polynomial of a global function field.

Syntax:

L := AlffLPolyRed(F);
L := AlffLPolyRed(F, "nocheck");

polynomial	L
F global function field	F

AlffLinearSeriesEnumElt

Return the current divisor in the series enumeration.

Syntax:

E := AlffLinearSeriesEnumElt(env);

alff divisor	E	current divisor
record	env	of enumeration data

AlffLinearSeriesEnumEnv

Prepare for enumeration of a linear series over a finite field.

Syntax:

env := AlffLinearSeriesEnumEnv(D, B);

record	env	of enumeration data
alff divisor	D
list	B	of v_1, \dots, v_l

AlffLinearSeriesEnumNext

Whether there is a next divisor in the linear series enumeration.

Syntax:

b := AlffLinearSeriesEnumNext(env);

bool	b	whether there is a next divisor
record	env	of enumeration data

AlffOrderAlff

Returns the algebraic function field of an algebraic function field order.

Syntax:

F := AlffOrderAlff(o);

algebraic function field	F
algebraic function field order	o

AlffOrderBasis

Returns a basis as elements of the given alff order with a denominator.

Syntax:

L := AlffOrderBasis(o);

list of algebraic elements	L
alff order	o

AlffOrderBasis

Returns a list containing the basis elements of an alff order.

Syntax:

B := AlffOrderBasis(o);

list	B	of basis elements
alff order	o

AlffOrderBasisValues

Returns some values depending on the B^*-values of the basis of an order of a global function field.

Syntax:

L := AlffOrderBasisValues(o);

list	L
global function field order	o

See also: Alff, AlffOrderL0, AlffOrderReduce

AlffOrderDedekindTest

Performs the Dedekind-Test on an equation order in an algebraic function field.

Syntax:

b := AlffOrderDedekindTest(o);
b := AlffOrderDedekindTest(o, g);

boolean	b
algebraic function field order	o
polynomial	g

AlffOrderDeg

Returns the degree of an alff order over its coefficient ring.

Syntax:

d := AlffOrderDeg(o);

integer	d	degree
alff order	o

AlffOrderDisc

Returns the discriminant of an algebraic function field order up to a unit.

Syntax:

d := AlffOrderDisc(o);

polynomial or quotient field element	d
algebraic function field order	o

AlffOrderEqFinite

Computes an equation order of the given algebraic function field.

Syntax:

o := AlffOrderEqFinite(F);

algebraic function field order	o
algebraic function field	F

See also: AlffOrderEqInfty, Alff, AlffElt

AlffOrderEqInfty

Computes an equation order of the given algebraic function field.

Syntax:

o := AlffOrderEqInfty(F);

algebraic function field order	o
algebraic function field	F

See also: AlffOrderEqFinite, Alff, AlffElt

AlffOrderIndex

Returns the index of an order.

Syntax:

d := AlffOrderIndex(o);

element	d	of the coefficient ring
algebraic function field order	o

AlffOrderIsFinite

Returns whether a function field order is finite.

Syntax:

b := AlffOrderIsFinite(o);

boolean	b
alff order	P

AlffOrderL0

Returns a list of 0-reduced basis elements of an order of a global function field with their B^*-values.

Syntax:

L := AlffOrderL0(o);

list	L
global function field order	o

AlffOrderMaxFinite

Computes a maximal order of an algebraic function field.

Syntax:

o := AlffOrderMaxFinite(F);

algebraic function field order	o
algebraic function field	F

AlffOrderMaxInfty

Computes a maximal order of an algebraic function field.

Syntax:

o := AlffOrderMaxInfty(F);

algebraic function field order	o
algebraic function field	F

AlffOrderMaximal

missing shortdoc

Syntax:

O := AlffOrderMaximal(o);

algebraic function field order	O
algebraic function field order	o

AlffOrderPoly

Returns the defining polynomial of the associated equation order of an algebraic function field.

Syntax:

f := AlffOrderPoly(o);

polynomial	f
algebraic function field order	o

AlffOrderReduce

Given an order of a global function field, the function returns an order with a 0-reduced basis.

Syntax:

o2 := AlffOrderReduce(o1);

global function field order	o2
global function field order	o1

AlffOrderTransformationMatrix

Returns a list containing information about the transformation from the alffsuborder of the given alfforder to the given alfforder.

Syntax:

L := AlffOrderTransformationMatrix(O);

list	L
alfforder	O

AlffPlaceAlff

Given a place this function returns the algebraic function field it is belonging to.

Syntax:

F := AlffPlaceAlff(P);

algebraic function field	F
alff place	P

AlffPlaceBeta

Return an inverse prime element for a place.

Syntax:

b := AlffPlaceBeta(P);

alff element	b
alff place	P

AlffPlaceDeg

Computes the degree of a place.

Syntax:

d := AlffPlaceDeg(P);

integer	d
alff place	P

AlffPlaceIdeal

missing shortdoc

Syntax:

I := AlffPlaceIdeal(P);

alff order ideal	I
alff place	P

AlffPlaceIsFinite

Returns whether a place is finite.

Syntax:

b := AlffPlaceIsFinite(P);

boolean	b
alff place	P

AlffPlaceMin

Returns the "minimum" of a place.

Syntax:

p := AlffPlaceMin(P);

prime polynomial or 1/T	p
alff place	P

AlffPlaceOrder

missing shortdoc

Syntax:

o := AlffPlaceOrder(P);

alff order	o
alff place	P

AlffPlacePrimeElt

missing shortdoc

Syntax:

u := AlffPlacePrimeElt(P);

alff element	u
alff place	P

AlffPlaceRam

Ramification of a place.

Syntax:

r := AlffPlaceRam(P);

integer	r
alff place	P

AlffPlaceRandom

Returns a random place of degree d.

Syntax:

p := AlffPlaceRandom(F,d);

alff place	p	of degree d
global function field	F
integer	d

AlffPlaceResDeg

Residue class field extension degree of a place.

Syntax:

f := AlffPlaceResDeg(P);

integer	f
alff place	P

AlffPlaceResField

Return the residue field of a place.

Syntax:

K := AlffPlaceResField(P);

field	K	residue field of P
alff place	P

AlffPlaceSplit

Decompose a rational function field place.

Syntax:

L := AlffPlaceSplit(F, p);

list	L	of places {frak P} above {frak p}
algebraic function field	F
prime polynomial or 1/T	p

AlffPlaceSplitType

Returns a list with ramification indices and relative degrees of all pairwise distinct places lying over a rational function field place.

Syntax:

L := AlffPlaceSplitType(F, p);

list	L
algebraic function field	F
polynomial or 1/T	p

AlffPlaceSplitType

Returns a list with ramification indices and relative degrees of all pairwise distinct places lying over a rational function field place.

Syntax:

L := AlffPlaceSplitType(F, p);

list	L
algebraic function field	F
polynomial or 1/T	p

AlffPlaces

Computes places of a global function field.

Syntax:

L := AlffPlaces(F,d);

list	L	of alff places
global function field	F
integer	d

AlffPlacesDegOne

Compute all places of degree one of a global function field.

Syntax:

L := AlffPlacesDegOne(F);

global function field	F
list	L	of alff places

AlffPlacesDegOne

Computes all places of degree one of a global function field.

Syntax:

L := AlffPlacesDegOne(F);

global function field	F
list	L	of alff places of degree 1

AlffPlacesDegOneNonSingFiniteNum

Computes the number of finite non singular places of degree one of a global function field's constant field extension of degree m.

Syntax:

b := AlffPlacesDegOneNonSingFiniteNum(F, m);

global function field	F
integers	b,m

AlffPlacesDegOneNum

Computes the number of places of degree one of a global function field's constant field extension.

Syntax:

N := AlffPlacesDegOneNum(F,m);

global function field	F
integer	N, m

AlffPlacesDegOneNum

Computes the number of places of degree one of a global function field's constant field extension.

Syntax:

N := AlffPlacesDegOneNum(F,m);

global function field	F
integer	N, m

AlffPlacesNonSpecial

Computes a non special system of g distinct places for a global function field F/k of genus g.

Syntax:

S := AlffPlacesNonSpecial(F);
S := AlffPlacesNonSpecial(F, d);

list	S	of alff places
global function field	F
integer	d	>= 1

AlffPlacesNum

Computes the number of places of a given degree of a global function field.

Syntax:

N := AlffPlacesNum(F, m);

global function field	F
integers	N, m

AlffPlacesNum

Computes the number of places of a given degree of a global function field.

Syntax:

N := AlffPlacesNum(F, m);

global function field	F
integers	N, m

AlffPoly

missing shortdoc

Syntax:

f := AlffPoly(F);

polynomial	f
algebraic function field	F

AlffPolyIrrIsSep

missing shortdoc

Syntax:

b := AlffPolyIrrIsSep(f);

polynomial in T and y over field	f
boolean	b

AlffPolyIsIrrSep

missing shortdoc

Syntax:

b := AlffPolyIsIrrSep(f);

polynomial in T and y over field	f
boolean	b

AlffPolyIsIrreducible

missing shortdoc

Syntax:

b := AlffPolyIsIrreducible(f);

polynomial in T and y over field	f
boolean	b

AlffPuiseuxCoeff

Returns the finite constant field of the Puiseux series field into which all completions of a global function field with respect to the infinite valuations can be embedded.

Syntax:

k := AlffPuiseuxCoeff(F);

finite field	k
global function field	F

AlffRamDivisor

Returns the ramification divisor of a divisor.

Syntax:

A := AlffRamDivisor(D);
A := AlffRamDivisor(F);

alff divisor	A	the ramification divisor of D
alff divisor	D
alff	F	equivalent to taking D = 0

AlffRegulator

Computes the regulator of a global function field's finite maximal order's unit group.

Syntax:

R := AlffRegulator(o);

integer	R
finite maximal order	o

AlffResFieldEltLift

Lifts a residue field element.

Syntax:

a := AlffResFieldEltLift(b, P);

alff element	a	in o_P
field element	b	value of a at P to be lifted
alff place	P

AlffRootParams

Sets/Displays the iteration depth of root expansions and the precision for series operations.

Syntax:

L := AlffRootParams(F);
L := AlffRootParams(F, n, m);

list	L
global function field	F
integer	n
integer	m

AlffRoots

Computes the Puiseux expansions in T^{-1/e} of the roots of the defining polynomial of a global function field F/FF_q(T).

Syntax:

L := AlffRoots(F);

list	L
global function field	F

AlffSUnits

Find the S-regulator and a basis for the free part of the S-units.

Syntax:

L := AlffSUnits( S [, "raw" ] );

list	L	the S-regulator and a basis of S-units
list	S	a list of places

AlffSerreBound

Computes the Serre bound of a global function field.

Syntax:

b := AlffSerreBound(F);
b := AlffSerreBound(q, g);

global function field	F
integer	q, g
integer	b

AlffSignature

Returns the signature of a global function field extension F/FF_q(T).

Syntax:

L := AlffSignature(F);

list	L
global function field	F

AlffTameInftyPlace

Returns whether the infinite place of k(T) is tamely ramified in the algebraic field extension F/k(T), or not.

Syntax:

b := AlffTameInftyPlace(F);

boolean	b
algebraic function field	F

AlffUnitRank

Returns the unit rank of a global function field extension F/FF_q(T).

Syntax:

r := AlffUnitRank(F);

integer	r
global function field	F

AlffUnitsFund

Computes s-1 fundamental units of a global function fields's finite maximal order.

Syntax:

L := AlffUnitsFund(o);

list	L
finite maximal order	o	of a global function field

AlffVarT

missing shortdoc

Syntax:

T := AlffVarT(F);

algebraic function field	F
polynomial	T

AlffVarY

missing shortdoc

Syntax:

y := AlffVarY(F);

algebraic function field	F
polynomial	y

AlffWeierstrassPlaces

Returns the Weierstra\ss{} places of a divisor.

Syntax:

L := AlffWeierstrassPlaces( D );
L := AlffWeierstrassPlaces( F );

list	L	containing the Weierstra\ss{} places
alff divisor	D
alff	F	equivalent to taking D = 0

AlffWronskian

Returns the Wronski matrix of a Riemann-Roch space.

Syntax:

W := AlffWronskian(D);
W := AlffWronskian(F);

list	W	list of rows
alff divisor	D	{cal L}(D) is computed
alff	F	equivalent to taking D = 0

AlffWronskianOrders

Returns the Wronski orders of a Riemann-Roch space.

Syntax:

W := AlffWronskianOrders(D);
W := AlffWronskianOrders(F);

list	W	list of rows
alff divisor	D	{cal L}(D) is computed
alff	F	equivalent to taking D = 0

ArcCos

Returns the inverse cosine of a number.

Syntax:

y := ArcCos(x);

complex	y
complex	x

ArcSin

Returns the inverse sine of a number.

Syntax:

y := ArcSin(x);

complex	y
complex	x

ArcTan

Returns the inverse tangens of a number.

Syntax:

y := ArcTan(x);

complex	y
complex	x

Arg

Returns the argument of a complex number.

Syntax:

phi := Arg(z);

real	phi
complex	z

BagRead

Reads a bag from an open file.

Syntax:

BagRead(file); BagRead("filename");

File

  file

BagWrite

Writes a bag to an open file.

Syntax:

BagWrite (file, arg [,arg]); BagWrite("filename", arg [,arg]);

File	file
any expression	arg

Bell

Sounds the terminal bell.

Syntax:

Bell();

Bernoulli

Returns a list of Bernoulli numbers.

Syntax:

bern := Bernoulli(n);

integer	n
list	bern

BernoulliMagma

Returns a list of Bernoulli numbers.

Syntax:

bern := BernoulliMagma(n);

integer	n
list	bern

Ceil

Returns the minimal integer greater than or equal to a number.

Syntax:

y := Ceil(x);

integer	y
real	x

CharPoly

Computes the characteristic polynomial of an algebraic element, a matrix or an alff element

Syntax:

p := CharPoly (a [,PA]);
p := CharPoly (a [,O]);
p := CharPoly( M );

polynomial	p
polynomial algebra	PA
suborder	O
algebraic element alff order element	a
matrix	M

CharToInt

Returns the ASCII code of a character.

Syntax:

i := CharToInt(c);

character	c
integer	i

Characteristic

Return the characteristic of a ring.

Syntax:

m := Characteristic(R);

integer	m
ring	R

CheckArgus

Handy tool to check the input to a kash function.

Syntax:

r = CheckArgus (hdCall,string);

r	int	number of type matching format string
hdCall	TypHandle	all arguments
string of format strings	t_string

ChineseRemainder

Chinese remainder for integers, ideals, and polynomials.

Syntax:

elt := ChineseRemainder(M,L);
elt := ChineseRemiander(LM);
elt := ChineseRemainder(m1, m2, a1, a2);

integer, polynommial, or order element	elt
list of elements (integers, polynomials, or order elements)	L
list of modules (integers, polynomials, or ideals)	M
list of pairs of elements and modules	LM
modules (integers, polynomials, or ideals)	m1,m2
elements (integers, polynomials, or order elements)	a1,a2

Close

Closes a file.

Syntax:

closed := Close(f);

File	f
boolean	closed

See also: BagRead, BagWrite, ECHOon, ECHOoff, FLDin, FLDout, LOFILES, Open

Coeff

Syntax:

a := Coeff(poly, i [,"x"]);

ring element	a
polynomial	poly
positive integer	i

ColorString

Defines a string for use in Print to get colorful output.

Syntax:

s := ColorString(l);

string	s
list of strings	l

Colors

Activate or deactivate color mode and customize colors.

Syntax:

Colors(flag);
Colors(s1, ..., sn);
Colors(L);

boolean	flag	switch color mode on or off
string	s1, ..., sn	resource strings
list	L	list of lists of resource strings

Comp

Creates a complex number.

Syntax:

z := Comp(a, b);

complex	z
real	a
real	b

ComplexGamma

Returns .

Syntax:

c := ComplexGamma(z);

complex	z
complex	c

Conj

Returns the conjugate complex number.

Syntax:

w := Conj(z);

complex	w
complex	z

Cos

Returns the cosine of a number.

Syntax:

y := Cos(x);

complex	y
complex	x

DbQuery

Performs a query in the database "kate"

Syntax:

DbQuery(query [, num])

string	query
integer	num

DedekindEta

Calculates the value of Dedekind's \eta-function.

Syntax:

u := DedekindEta(z);

complex	u
complex	z	complex number with Im(z)>0

Den

Returns the denominator of an object.

Syntax:

d := Den( a );
d := Den(a, "rep");
d := Den( q );
I := Den( I );

integer	d
quotient field element or polynomial	d
rational	q
algebraic element	a
algebraic function field order element	a
quotient field element or polynomial	q
ideal	I

Disc

Computes the discriminant of a polynomial, an order, an algebraic element, an alff order or a lattice.

Syntax:

d := Disc(f);
d := Disc(o);
d := Disc(l);

ring element	d
polynomial	f
order alff order	o
lattice	l

ECHOoff

Switches back to normal behaviour after ECHOon.

Syntax:

ECHOoff();

ECHOon

Switches the output of stdout to a file.

Syntax:

ECHOon ( name \| file [, arg] );

string	name	filename to write to
file	file	open for writing
expression	arg	should produce an output to stdout

EccDecrypt

Decrypt a message that was encrypted using the ElGamal public key cryptosystem and a subgroup of the group of an elliptic curve.

Syntax:

P := EccDecrypt(K,E,a,M);

finite field	K
list	E
integer	a
list	M
list	P

EccEncrypt

Encrypt a message (point) using the ElGamal public key cryptosystem and a subgroup of the group of an elliptic curve.

Syntax:

M := EccEncrypt(K,E,B,aB,k,P);

finite field	K
list	E
list	B
list	aB
integer	k
list	P
list	M

EccInit

Initializes a representation of an elliptic curve for the Ecc-package.

Syntax:

ec := EccInit(p,a,b);
ec := EccInit(p,li);
ec := EccInit(fli);

elliptic curve	ec
integer	a
integer	b
integer	p
list of integers	li
list of finite field elements	fli

EccIntPointMult

Scalar multiplication of a point on an elliptic curve.

Syntax:

nP := EccIntPointMult(K,E,n,P);

finite field	K
list	E
integer	n
list	P
list	nP

EccKangaroo

Computes the number of places of an elliptic curve, given by the Weierstrass form over a prime finite field F_p.

Syntax:

NP:=EccNumberOfPoints(p | F,a,b);
NP:=EccNumberOfPoints(ec);

int or finite field element	a
int or finite field element	b
int	p
finite field	F
list	ec

EccPointIsOnCurve

Checks whether a point is on a given elliptic curve.

Syntax:

b := EccPointIsOnCurve(K,ec,point);

finite field	K
list	ec
list	point
boolean	b

EccPointsAdd

Returns the sum of two points on an elliptic curve over a finite field.

Syntax:

sum := EccPointsAdd(K,ec,P1,P2);

finite field	K
list	E
list	P1
list	P2
list	sum

EccRandomPoint

Returns a random point on a given elliptic curve.

Syntax:

P := EccRandomPoint(F,E);

point on the elliptic curve E	P
finite field	F
elliptic curve	E

Ei

This function computes an approximation of the integral from -\infty to x of exp(t)/t.

Syntax:

a := Ei(b);

real	a
real	b

EisensteinSeries

Returns the value of the non-holomorphic Eisensteinseries.

Syntax:

c := EisensteinSeries(tau,s,a1,a2,q,N);

complex	tau
real	s
integer	a1
integer	a2
integer	q
positive integer	N

Elt

Creates an algebraic number.

Syntax:

a := Elt(O,L);
a := Elt(O,L/d);
a := Elt(O,h/d);
a := Elt(O,h);

algebraic element	a
order	O
list	L	of coefficients
integer	h
integer	d

EltAbs

missing shortdoc

Syntax:

v := EltAbs (a);

real matrix	v
algebraic element	a

EltAbsLogHeight

Computes the absolute logarithmic height of an algebraic number.

Syntax:

h := EltAbsLogHeight(a);

real	h
algebraic element	a

EltApproximation

Returns an element with certain valuations at prime ideals.

Syntax:

E := EltApproximation(P, L);

list	P	of distinct prime ideals over the same order
list	L	of small integers
algebraic element	E

EltAutomorphism

Applies an automorphism to an algebraic number.

Syntax:

L := EltAutomorphism (a);
c := EltAutomorphism (a,i);

list	L
algebraic element	c
algebraic element	a
integer	i

EltCharPoly

Characteristic polynomial of an algebraic element over a subfield.

Syntax:

p := EltCharPoly (a [, PA]);
p := EltCharPoly (a [, O]);

polynomial	p
polynomial algebra	PA
suborder	O
algebraic element	a

EltCon

Computes the conjugates of an algebraic number.

Syntax:

v := EltCon(alpha);
c := EltCon(alpha,i);

complex matrix	v
complex number	c
algebraic element	alpha
integer	i

EltDen

Returns the denominator of an algebraic element.

Syntax:

d := EltDen(a);
d := EltDen(a, "rep");

integer	d
algebraic element	a

EltDivisors

Computes all divisors of an algebraic integer.

Syntax:

L := EltDivisors (a);

list	L
algebraic integer	a

EltExcepUnitOrbit

Computes the orbit of an exceptional unit.

Syntax:

L := EltExcepUnitOrbit(epsilon);

list	L
algebraic element	epsilon

EltIdealReduce

Returns a canonical representative modulo an ideal.

Syntax:

b := EltIdealReduce(a,I); same as b := EltIdealReduce(a,I,HNF);
b := EltIdealReduce(a,I,LLL);
b := EltIdealReduce(a,I,INTEGRAL);
b := EltIdealReduce(a,I,HNF_POS);
b := EltIdealReduce(a,I,INTEGRAL_POS);

algebraic element	b
algebraic element	a
Ideal	I
interpreted as strings	HNF, LLL, INTEGRAL, HNF_POS, INTEGRAL_POS

EltIndex

Computes the index of an equation suborder of a given order

Syntax:

index := EltIndex (alpha [,Z]);

integer	index
algebraic integer	alpha
ring of integers	Z

EltIsInIdeal

missing shortdoc

Syntax:

b := EltIsInIdeal (alpha,a);

boolean	b
algebraic element	alpha
ideal	a
	INDES test on; element of an ideal
	check for; element of an ideal
	algebraic number; element of an ideal test
	element of an ideal test

EltIsInt

missing shortdoc

Syntax:

b := EltIsInt(a);

boolean	b
algebraic element	a

Tests whether the element is an algebraic integer resp. an element of the order. The second parameter "order" checks, if the element is integral in the mathematical sence, e.g. if it has a denominator.

Syntax:

b := EltIsIntegral(a);
b := EltIsIntegral(a, "order");

boolean	b
algebraic element	a

EltIsPrimitive

Checks whether an algebraic element is primitive.

Syntax:

b := EltIsPrimitive (a);

boolean	b
algebraic element	a

EltListAbsDisc

Discriminant of a module.

Syntax:

d := EltListAbsDisc (L);

list	L	of algebraic elements
integer	d

EltListToMat

Converts a list of algebraic numbers into a matrix of their coefficients.

Syntax:

M := EltListToMat(L);

matrix	M	matrix over the coefficient order of the elements of L
list	L	of algebraic numbers

EltLogs

Returns a matrix v with the logarithms of the absolute values of the conjugates the algebraic number a. Matrix v has length r|1+r|2.

Syntax:

v := EltLogs(a);

real matrix	v
algebraic element	a

EltMatToList

Converts a matrix of algebraic numbers over a maximal order o into a dynamic array of elements of the relative order O.

Syntax:

L := EltMatToList(O, M);

list	L	of algebraic numbers over O
matrix	M	of algebraic numbers
order	O	relative order over the coefficient order of the elements of the matrix

EltMinPoly

Minimal polynomial of an algebraic element over a subfield.

Syntax:

p := EltMinPoly (a [, PA]);
p := EltMinPoly (a [, O]);

polynomial	p
polynomial algebra	PA
suborder	O
algebraic element	a

EltMinkowski

Returns the image of an algebraic element under the Minkowski map.

Syntax:

v := EltMinkowski(a);

real matrix	v
algebraic element	a

EltMove

Returns an algebraic element in a different representation.

Syntax:

b := EltMove(a,S);

algebraic element or lattice element	b
algebraic element or lattice element	a
order or lattice	S

EltNewtonLift

Lifts an algebraic element with the Newton lifting method.

Syntax:

alpha := EltNewtonLift(o, a, f, p, k);

algebraic element	alpha
order	o
algebraic element	a
polynomial	f
integer	p
integer	k

EltNorm

Returns the norm of an algebraic element in a given order.

Syntax:

n := EltNorm(a [,o]);

rational number or algebraic element	n
algebraic element	a
order	o

EltNumberReduce

Returns a canonical representative modulo an integer or rational.

Syntax:

b := EltNumberReduce(a,m);

integer or rational number	m
algebraic element	a
algebraic element	b

EltOrder

Returns the order of an algebraic element.

Syntax:

o := EltOrder(alpha);

order	o
algebraic element or integer	alpha

EltPowerMod

Computes a^ {pow} \bmod M for an algebraic element.

Syntax:

e := EltPowerMod(a, pow, M);

algebaric element	e, a
integer	`{pow`}
integer	M

EltPowerProduct

Returns an algeraic element \beta with the power product of the algebraic elements in Alpha and the corresponding exponents in Expons. The matrices Alpha and Expons must have the same length and only one row.

Syntax:

beta := EltPowerProduct(o,Alpha,Expons);

algebraic element	beta
order	o
matrix of algebraic elements	Alpha
matrix of integers	Expons

EltRayResidueRingRep

Returns a representive of the class of an algebraic element in the multiplicative group of the ray residue ring.

Syntax:

r := EltRayResidueRingRep(elt,m0,minf);

matrix	r
algebraic element	elt
ideal	m0
list	minf	of integers/infinite primes

EltReconstruct

Lifts an element from a modulo m approximation to an element with rational coefficients

Syntax:

alpha := EltReconstruct (gamma, m);

false or algebraic element	alpha
algebraic element	gamma
integer	m

EltRepMat

A representation matrix of an algebraic element over o together with a suitable denominator.

Syntax:

L := EltRepMat(a[,o]);

list	L
algebraic element	a
order	o	must be a direct suborder of EltOrder (a)

EltRoot

Computes a root of an algebraic element.

Syntax:

beta := EltRoot(alpha,m);
beta := EltRoot(alpha,m [,"enum"|"mode"]);
beta := EltRoot(alpha,m ,"enum", mode);

false or algebraic element	beta
algebraic element	alpha
small integer	m
small integer or string	mode
string	enum

EltSimplify

Returns a in simplified representation.

Syntax:

e := EltSimplify(a);

integer or algebaric element	e
integer or algebraic element	a

EltToFFE

Returns the class of an algebraic element viewed as an element of a finite field.

Syntax:

f := EltToFFE( a, p);

integer or algebraic element	a
prime ideal	p
finite field element or integer	f	interpreted as finite field element

EltToList

Returns the coefficient list of the algebraic element a.

Syntax:

L := EltToList(a);

algebraic element	a
list	L	of integers or algebraic elements

EltTrace

Returns the trace of an algebraic element. The trace is contained in the coefficient ring of the defining order or in the given order o.

Syntax:

t := EltTrace (a [,o]);

rational number or algebraic element	t
algebraic element	a
order	o

EltUnitDecompose

Returns the decomposition of an unit with respect to the computed system of fundamental units of a maximal order.

Syntax:

F := EltUnitDecompose(u);
L := EltUnitDecompose(u,"expons");

list	F
list	L
algebraic element	u

EltValuation

Compute the valuation of an algebraic element or an integer.

Syntax:

val := EltValuation(P, alpha);

	val}{integer}{returned value.
	P}{ideal}{a prime ideal.
	alpha}{algebraic element or integer}{number to evaluate.

EulerGamma

Returns the value of the Euler constant \gamma.

Syntax:

y := EulerGamma();

real

Eval

Evaluates a polynomial at a value.

Syntax:

y := Eval(f, s);

	y
polynomial	f
	s

Exp

Returns the exponential of a number.

Syntax:

y := Exp(x);

complex	y
complex	x

FF

Creates a finite field.

Syntax:

F := FF(p);
F := FF(p, d);
F := FF(f);

finite field	F
integer	p
integer	d
polynomial	f

See also: FFElt, FFGenerator, FFPrimitiveElt

FFCreate

Creates a finite field.

Syntax:

FFCreate(O,p,k)

order	O
integer	p
integer	k

FFEToElt

Returns a canonical representative of a finite field element viewed as a representative of a class of algebraic numbers.

Syntax:

a := FFEToElt( f, p);

finite field element	f
ideal	p	must be prime
algebraic number or integer	a	interpreted as algebraic number

FFElt

Creates an element of a finite field.

Syntax:

a := FFElt(F, n);

finite field element	a
finite field	F
integer	n

FFEltFF

Gives the finite field corresponding to the element.

Syntax:

F := FFEltFF(a);

finite field element	a
finite field	F

FFEltIsZero

Returns true iff the argument is of type "KANT finite field elt" AND zero.

Syntax:

b := FFEltIsZero(a);

boolean	b
	a

FFEltLog

Discrete logarithm for finite fields.

Syntax:

d := FFEltLog(a);

finite field element	a
integer	d

See also: FFPrimitiveElt

FFEltMinPoly

Minimal polynomial of finite field elements.

Syntax:

p := FFEltMinPoly(a);
p := FFEltMinPoly(a, J);
p := FFEltMinPoly(a, Jx);

polynomial	p
finite field element	a
finite field	J	subfield of the finite field of a
polynomial algebra in `x` over `J`	Jx

See also: FFPrimitiveElt

FFEltMove

Move finite field element into another finite field if possible.

Syntax:

b := FFEltMove(a, J);

finite field element	b
finite field element	a
finite field	J

See also: FFPrimitiveElt

FFEltNorm

Norm of finite field elements.

Syntax:

p := FFEltNorm(a);
p := FFEltNorm(a, J);

polynomial	p
finite field element	a
finite field	J	subfield of the finite field of a

See also: FFPrimitiveElt

FFEltRoot

Syntax:

b := FFEltRoot(a, n);

finite field element	b
finite field element	a
small integer	n

FFEltToInt

Converts an element of a finite prime field to the corresponding integer.

Syntax:

i := FFEltToInt(b);

finite field element	b
integer	i

FFEltToList

Returns a basis representation of a finite field element.

Syntax:

L := FFEltToList(b);
L := FFEltToList(b, k);

finite field element	b
subfield	k
list	L

FFEltTrace

Trace of finite field elements.

Syntax:

p := FFEltTrace(a);
p := FFEltTrace(a, J);

polynomial	p
finite field element	a
finite field	J	subfield of the finite field of a

See also: FFPrimitiveElt

FFEmbed

Embeds one finite field into another.

Syntax:

FFEmbed(F1, F2);
FFEmbed(F1, F2, b);

finite field	F1
finite field	F2
finite field element	b

FFGenerator

Returns a generator of a finite field.

Syntax:

a := FFGenerator(F);
a := FFGenerator(F, k);

finite field element	a
finite field	F
subfield of F	k

FFPrimitiveElt

Returns a primitive element of a finite field.

Syntax:

w := FFPrimitiveElt(F);

finite field element	w
finite field	F

FFSize

Gives the size of the finite field FF_q.

Syntax:

L := FFSize(F);

list	L	of p and d with p^d = q.
finite field	F

FLDin

Reads an order in the FLD format.

Syntax:

o := FLDin(name [, n]);
o := FLDin();
o := FLDin(f [, n]);

file	f	open for reading
string	name	filename
order	o
integer	b	the number of fields to skip. If given, the number
	of fields is returned.

FLDin_DB

Reads an order in the FLD format for database "kate" without fundamental units and only number of generators of the class group and their orders

Syntax:

o := FLDin(name [, n]);
o := FLDin();
o := FLDin(f [, n]);

file	f	open for reading
string	name	filename
order	o
integer	b	the number of fields to skip. If given, the number
	of fields is returned.

FLDout

Writes an order using the FLD format.

Syntax:

FLDout(o [, name | file]);

Order	o
String	name	filename to use
File	file	opened for writing

See also: Open, Close, LOFILES, FLDin, ECHOon, ECHOoff

Factor

Returns the factorization of the given argument.

Syntax:

F := Factor(d);
F := Factor(f);
F := Factor(f, p);
F := Factor(a);
F := Factor(alpha);
F := Factor(O, d);
F := Factor(O, a);
F := Factor(O, alpha);

list	F
prime	p
integer	d
polynomial	f
ideal	a
algebraic element	alpha
order	O

FilePosition

Sets or reads the file position, allowing thus random access of FLD files.

Syntax:

o := FilePosition(file [, pos ]);

file

open for reading

FindMaximalCentralField

Finds the maximal factorgroup where a list of automorphisms acts trivial.

Syntax:

sg := FindMaximalCentralField(o | I [, inf][, aut]);
sg := FindMaximalCentralField(G [, aut]);

Matrix	sg	describing the aditional relations
order	o
integral ideal	I
list	inf	of infinite places
AbelianGroup	G	must be from RayClassGroupToAbelianGroup
list	aut	of automorphisms of `o`, if omitted `OrderAutomorphisms(o, [])` is used.

FindQuotientOfShapeEnumInit

Initializes an environment to enumerate certain subgroups

Syntax:

s := FindQuotientOfShapeEnumInit(G, L);

record	s
AbelianGroup	G
list	L	of integers describing the shape of the subgroup

FindQuotientOfShapeEnumNext

Steps through the subgroups.

Syntax:

flag := FindQuotientOfShapeEnumNext(s [, 1]);

boolean	flag
record	s	generated by `FindQuotientOfShapeEnumInit`
if present find subgroups instead of quotients.	1

Floor

Returns the maximal integer less than or equal to a given number.

Syntax:

y := Floor(x);

integer	y
real	x

GPin

Reads an order in the pari/gp format.

Syntax:

o := GPin(name);
o := GPin();
o := GPin(f);

file	f	file opened for reading
string	name	filename
order	o
	**

Galois

Computation of Galois groups.

Syntax:

Galois ( f [,p] [, "fast"] );
Galois ( o [,p] [, "fast"] );

Galois ( f , "complex" [, n] );
Galois ( o , "complex" [, n] );

GaloisT ( f [,p] [, "fast"] );
GaloisT ( o [,p] [, "fast"] );

GaloisT ( f , "complex" [, n] );
GaloisT ( o , "complex" [, n] );

polynomial	f
order	o
prime number	p
positive integer	n	precision

GaloisBlocks

Excluding unpossible Galois groups by blocks.

Syntax:

GaloisBlocks(o);
GaloisBlocks(o, true);

order

GaloisGlobals

Prints the current settings in the Galois group computation.

Syntax:

GaloisGlobals(o);

order

See also: Galois, GaloisTree, GaloisRoots

GaloisGroupKnown

This function returns the Galois group if it is already set in the order or the algebraic function field. If the Galois group is not set false will be returned.

Syntax:

b:= GaloisGroupKnown(K);

int or false	b
order or algebraic function field	K

GaloisGroupOrder

Returns the order of the transitive permutation group.

Syntax:

GaloisGroupOrder(n, k);

integer	n	representing degree
integer	k	representing group in T-notation

See also: Galois

GaloisGroupSet

Sets the number of the Galois group in the order or algebraic function field.

Syntax:

GaloisGroupSet(K, G);

order or algebraic function field	K
int	G

GaloisGroupsPossible

Handling of possible Galois groups.

Syntax:

L := GaloisGroupsPossible(o);
GaloisGroupsPossible(o, G, flag);
GaloisGroupsPossible(o, L, flag);

order	o
integer	G	representing transitive group in T-notation
list	L	of integers G
boolean	flag	whether to add or remove groups

GaloisMSetPol

Computing of a polynomial of degree {n choose k} which roots are products of k distinct roots of f.

Syntax:

g:= GaloisMSetPol(f, k);

int	k	must be positive
polynomial	f, g

GaloisMSumPol

Computing of a polynomial of degree {n choose k} which roots are sums of k distinct roots of f.

Syntax:

g:= GaloisMSumPol(f, k);

int	k	must be positive
polynomial	f,g

GaloisMissionS

Returns all non-trivial subfields of given degree m. If no m is specified, all subfields are calculated.

Syntax:

L := GaloisMissionS(o);
L := GaloisMissionS(o, m);

list	L	list of suborders
order	O	the given order
small integer	m	the prescribed degree of subfields

GaloisModulo

Excluding impossible Galois groups by cycle types.

Syntax:

GaloisModulo(o, b);

order	o
integer	b	bound

GaloisNumberToName

Returns name of transitive permutation group.

Syntax:

GaloisNumberToName(n, k);

integer	n	representing degree
integer	k	representing group in T-notation

See also: Galois

GaloisRing

missing shortdoc

Syntax:

L := GaloisRing(o);

order	o
list	L	of either the p-adic order, the prime (ideal) p and theexponent k or the complex field and the precesion

See also: Galois

GaloisRoots

Returns current root ordering.

Syntax:

L := GaloisRoots(o);
L := GaloisRoots(o,k);

order	o
int	k
list	L	of list of roots and permutation

See also: Galois

GaloisSymb

Computation of Galois groups

Syntax:

G := GaloisSymb(o);
G := GaloisSymb(F);
G := GaloisSymb(f);

string	G
order	o
algebraic function field	F
polynomial	f

See also: Galois

GaloisSymb

Computation of the Galois group.

Syntax:

G := GaloisSymb(o|F|f);

string	G
order	o
algebraic function field	F
polynomial	f

See also: Galois

GaloisSymbT

Computation of Galois groups

Syntax:

G := GaloisSymbT(o);
G := GaloisSymbT(F);
G := GaloisSymbT(f);

int	G
order	o
algebraic function field	F
polynomial	f

See also: Galois

GaloisSymbT

Computation of the Galois group.

Syntax:

G := GaloisSymbT(o|F|f);

int	G
order	o
algebraic function field	F
polynomial	f

See also: Galois

GaloisT

Computation of Galois groups.

Syntax:

GaloisTree

Output of possible Galois groups as subgroup lattice.

Syntax:

L := GaloisTree(o);

order	o
list	L	list of two lists of integers

GaloisTreeRoots

Return the roots of the tree of possible Galois groups.

Syntax:

L := GaloisTreeRoots(o);

order	o
list	L	list of two lists of integers

GaloisTwoSequencePol

Computing of a polynomial of degree ncdot(n-1) which roots are products of two distinct roots of f.

Syntax:

g:= GaloisTwoSequencePol(f);

polynomial

  f, g

Gamma

Returns the value of the \Gamma-function.

Syntax:

y := Gamma(x);

real	y
real	x

Gcd

Returns greatest common divisor of list of arguments.

Syntax:

gcd := Gcd(L);
gcd := Gcd(a, b);

integer or polynomial or ideal	gcd
list of integers of polynomials or ideals	L
integer or polynomial or ideal	a
integer or polynomial or ideal	b

GetEnvironment

Returns the current value of the environment variable name (\name).

Syntax:

st := GetEnvironment(name);

string	st
string	name

HermiteUpperBound

Returns a upper bound for Hermite's constant \gamma_ n^ n (according to Blichfeldt Blich).

Syntax:

y := HermiteUpperBound(n);

real	y
integer	n

HurwitzZeta

Returns the value of the Hurwitz Zeta-function.

Syntax:

c := HurwitzZeta(s,v,m0);

complex	s
real	v
integer	m0
complex	c

Ideal

Creates an ideal defined by the arguments.

Syntax:

I := Ideal(e);
I := Ideal(o, n);
I := Ideal(e, f);
I := Ideal(n, e);
I := Ideal(o, M, d);
I := Ideal(o, MO);

Ideal	I
algebraic element	e
algebraic element	f
order over Z	o
matrix	M	of integers
integer	d	denominator of M
integer	n
module	MO	over the coefficient order of o

Ideal2EltAssure

The 2-element representation of the ideal is computed if it is not given yet.

Syntax:

Ideal2EltAssure(I);

ideal

Ideal2EltIntAssure

The 2-element representation where the first generator is a rational integer of the ideal is computed if it is not given yet.

Syntax:

Ideal2EltIntAssure(I);

ideal

Ideal2EltKnown

Returns true if the ideal is given in two element representation.

Syntax:

b := Ideal2EltKnown(I);

Boolean	b
ideal	I

See also: IdealGenerators

Ideal2EltNormalAssure

Calculates a 2 element normal representation of an ideal.

Syntax:

Ideal2EltNormalAssure(I);

ideal

See also: IdealGenerators

Ideal2EltNormalKnown

True if the ideal is given with normal two elements representation.

Syntax:

b := Ideal2EltNormalKnown(I);

Boolean	b
ideal	I

See also: IdealGenerators

IdealAutomorphism

Applies an automorphism to an ideal.

Syntax:

B := IdealAutomorphism (A,i);

ideal	B
ideal	A
integer	i

IdealBasis

Returns the basis of an ideal.

Syntax:

M := IdealBasis(I);

ideal	I
list	M	two elements, the denominator and the representation matrix

See also: IdealGenerators

IdealBasisKnown

Returns true if the ideal is given via a Z basis.

Syntax:

b := IdealBasisKnown(I);

Boolean	b
ideal	I

IdealBasisLowerHNF

The basis of the ideal transformed in lower HNF

Syntax:

M:=IdealBasisLowerHNF(I);

list	M	two elements, the denominator and the basis matrix
ideal	I

IdealBasisUpperHNF

The basis of the ideal transformed in upper HNF.

Syntax:

M := IdealBasisUpperHNF(I);

list	M	two elements, the denominator and the basis matrix
ideal	I

IdealChineseRemainder

Chinese Remainder Theorem for number fields

Syntax:

beta := IdealChineseRemainder(a1, a2, alpha1, alpha2)

ideals	a1, a2
algebraic numbers	alpha1, alpha2
algebraic number	beta

IdealClassRep

Computes a representation of the ideal class over the cyclic generators of the class group.

Syntax:

L := IdealClassRep(I);
L := IdealClassRep(I, "gen");

list	L
ideal	I

IdealCollection

Solves a special equation.

Syntax:

M := IdealCollection(I1,I2);

list	M	two lists of two elements of the order
ideals	I1,I2	integral, over a maximal order

IdealDegree

Calculates the degree of inertia of a prime ideal.

Syntax:

d := IdealDegree(I);

integer	d
ideal	I	must be prime

IdealDen

The denominator of the ideal.

Syntax:

d := IdealDen(I);

ideal	I
integer	d

IdealDivisors

Computes all divisors of an ideal.

Syntax:

L := IdealDivisors (A);

list	L
ideal	A

IdealFactor

Returns the factorization of the given ideal over a maximal order.

Syntax:

F := IdealFactor(a);

list	F
ideal	a

See also: Factor

IdealGen

missing shortdoc

Syntax:

g := IdealGen(I, i);

algebraic element	g
ideal	I
small integer	i	{\in{1,2}}

IdealGenerators

Returns the 2 generators of the 2-element-representation of the ideal.

Syntax:

L := IdealGenerators(I);

ideal	I
list	L	of two algebraic numbers

IdealIdempotents

Returns elements of the comaximal ideals which sum to 1.

Syntax:

E := IdealIdempotents(L)

list	L	of integral ideals over the same order
false or list	E	of algebraic elements over the same order

IdealImprove

Improves the generators of the ideal given in two element representation.

Syntax:

I1:= IdealImprove(I2);

ideal

  I1, I2

IdealIntegrity

Checks the ideals for various inconsistencies

Syntax:

errcount = IdealIntegrity(I);

ideal	I
small integer	errcount	the number of detected errors

IdealIsIntegral

Checks if a (fractional) ideal is an integral (or true) ideal.

Syntax:

b := IdealIsIntegral(I);

boolean	b
Ideal	I

IdealIsPrime

Checks whether an ideal is a prime ideal. Returns true or false.

Syntax:

b := IdealIsPrime(I);

boolean	b
ideal	I	integral

IdealIsPrincipal

Returns a principal generator of an ideal, if it is a principal ideal and false otherwise.

Syntax:

g := IdealIsPrincipal(I);
g := IdealIsPrincipal(I, classgroup);

ideal	I
false or algebraic number	g	principal generator

IdealLLL

Creates an ideal with LLL-reduced real basis

Syntax:

I := IdealLLL(a);

ideal

  I, a

IdealLcm

Least common multiplier of 2 ideals.

Syntax:

I:=IdealLcm(I1, I2);

Ideals

  I, I1, I2

IdealLowerHNFTrans

the transformation matrix from the original basis to the basis in lower HNF

Syntax:

M:=IdealLowerHNFTrans(I);

matrix	M
ideal	I

IdealMakeCoprime

missing shortdoc

Syntax:

c := IdealMakeCoprime(A,B));

ideal	A,B
algebraic number	c

IdealMakeInvCoprime

Subroutine of OrderRelNfColl

Syntax:

a := IdealMakeInvCoprime(I1,I2);

ideals	A, B	must be integral
algebraic number	a

IdealMin

The minimum of the ideal.

Syntax:

m:=IdealMin(I);

integer	m
ideal	I

IdealMove

Embeds one or more ideals into a given order.

Syntax:

I2 := IdealMove(I1, o);

same type as I1	I2
ideal \| list of ideals	I1
order	o

IdealNorm

Returns the norm of an ideal.

Syntax:

n := IdealNorm(I);

ideal	I
rational number	n

IdealOrder

This function returns the order of an ideal.

Syntax:

O := IdealOrder(I);

Order	O
Ideal	I

IdealPrimeCountInit

Initializes an environment for enumerating prime ideals

Syntax:

s := IdealPrimeCountInit(o[, m);

record	s
order	o
integer	m	only primes coprime to `m` are used.

IdealPrimeCountNext

Enumerates the next prime ideal.

Syntax:

p := IdealPrimeCountNext(s [, 1]);

prime ideal	p
record	s	from IdealPrimeCountInit
if present, p will be a pair [p, f] with p a prime of the coefficient order and `f` the degree of `p`	1

IdealPrimeElt

primitive element of an ideal

Syntax:

e:=IdealPrimeElt(I);

algebraic number	e
Ideal	I

IdealRadical

Computes a radical.

Syntax:

r := IdealRadical (A, O);

order	O
ideal	A
ideal	r

IdealRamIndex

Calculates the ramification index of a prime ideal.

Syntax:

d := IdealRamIndex(I);

ideal	I	must be prime
integer	d

IdealRayClassRep

Returns a representive of the class of an ideal in the ray class group.

Syntax:

r := IdealRayClassRep(I,m0,minf);

matrix	r
ideal	I
ideal	m0
list	minf	of integers/infinite primes

IdealRemainderSet

Computes a remainder system of an ideal

Syntax:

L:= IdealRemainderSet(I);

list	L
ideal	I

IdealResidueField

Returns the finite field defined by a prime ideal.

Syntax:

K := IdealResidueField(p);

ideal	p
finite field	K

IdealResidueFieldIsomorphism

Calculates the isomorphism between the residue fields of two prime ideals.

Syntax:

alpha := IdealResidueFieldIsomorphism(a,b);

algebraic element	alpha
ideal	a
ideal	b

IdealRingOfMultiplicators

Computes the ring of multiplicators of an ideal.

Syntax:

o := IdealRingOfMultiplicators (A);

order	o
ideal	A

IdealUpperHNFTrans

the transformation matrix from the original basis to the basis in upper HNF

Syntax:

M:=IdealUpperHNFTrans(I);

matrix	M
ideal	I

IdealValuation

Computes the valuation of an ideal at a prime ideal.

Syntax:

val := IdealValuation(p, I);

ideal	P	must be prime
ideal	I
integer	val

IdealWithNorm

Computed all ideals of o of a given norm n.

Syntax:

L := IdealWithNorm(n,o);

List	L
integer	n	>1
MaximalOrder	o

IdemLift

Lifts an idempotent.

Syntax:

alpha:=IdemLift(a, p, k);

algebraic element	alpha
algebraic element	a
integer	p
integer	k

Im

Returns the imaginary part of a complex number.

Syntax:

a := Im(z);

real	a
complex	z

ImQuadFormCreate

Generates a quadratic form.

Syntax:

g:=ImQuadFormCreate(D,p);
g:=ImQuadFormCreate(D,[a,b,c]);

ImQuadHilbert

Determines the Hilbert class field of an imaginary quadratic field.

Syntax:

O := ImQuadHilbert(o [,repr] [,"roots"] );

order	o	imaginary quadratic field
list	repr	representants for the ideal classes of o
order	O	Hilbert class field

ImQuadRayField

Determines the ray class field modulo ideal f over an imaginary quadratic field.

Syntax:

O := ImQuadRayField( f [,"char"  |  "field"  |  "maxord"] );

ideal	f	integral ideal in imaginary quadratic field
order	O	ray class field modulo `f`

Index

Computes the index of an algebraic element, an order or an alff order

Syntax:

I := Index (a);
I := Index (o);

coefficient ring element	I
algebraic element	a
order alff order	o

InftyGcd

Computes a greatest common divisor with respect to the degree valuation.

Syntax:

c := InftyGcd(a, b);

quotient field elements

  a,b,c

See also: InftyVal, InftyQuotRem, InftyLcm

InftyLcm

Computes a least common multiple with respect to the degree valuation.

Syntax:

c := InftyLcm(a, b);

quotient field elements

  a,b,c

See also: InftyVal, InftyQuotRem, InftyGcd

InftyQuotRem

Computes quotient and remainder with respect to the degree valuation.

Syntax:

L := InftyQuotRem(a, b);

list	L	of q and r
quotient field elements	a,b

InftyVal

Returns the degree valuation of a rational function or a Puiseux series defined over a finite field.

Syntax:

n := InftyVal(a);

integer	n
quotient field element	a	of FF_q(x)

See also: InftyQuotRem, InftyGcd, InftyLcm, AlffRoots

Insert

Insert an element into a list at a given position.

Syntax:

Insert(L, a, pos);

list	L
arbitrary object	a
integer	pos

IntDivisors

Returns a sorted list of all non-negative divisors of a rational integer.

Syntax:

L := IntDivisors(d);

list	L
integer	d

IntEulerPhi

Computes the Euler-\phi function of n, i.e. the number of coprime integers less than n.

Syntax:

a := IntEulerPhi(n);

integer	a
integer	n

IntFactor

Syntax:

F := IntFactor(d);

list	F
integer	d

See also: Factor

IntGcd

Returns the non-negative greatest common divisor of two integers.

Syntax:

gcd := IntGcd (a1,a2);

integer	gcd
integer	a1
integer	a2

IntIsPrime

Returns true iff the argument is a rational prime.

Syntax:

b := IntIsPrime(n)

boolean	b
integer	n

IntIsSquare

Returns true iff the argument is a square.

Syntax:

b := IntIsSquare(n)

boolean	b
integer	n

IntLcm

Returns the least positve common multiplier of two integers and zero if one of both is zero.

Syntax:

c := IntLcm(a, b);

integer	c
integer	a
integer	b

IntMoebiusMy

Computes the Moebius \mu-function.

Syntax:

a := IntMoebiusMy(n);

integers

  a, n

IntMoebiusMy

Computes the Moebius \mu-function.

Syntax:

a := IntMoebiusMy(n);

integers

  a, n

IntPowerMod

Returns the power of a given integer modulo an integer.

Syntax:

y := IntPowerMod(x, n, m);

integer	y
integer	x
integer	n
integer	m

IntPrimeDivisors

Returns a sorted list of all prime divisors of a rational integer.

Syntax:

L := IntPrimeDivisors(d);

list	L
integer	d

IntQuo

Returns the integer quotient of two integers.

Syntax:

q := IntQuo(a, b);

integer	q
integer	a
integer	b

IntRandomBits

Returns a random integerin the range 0 to 2^n-1. n must be >= 0.

Syntax:

y := IntRandomBits(n);

integer	y
small integer	n
	H

IntRoot

Returns rfloor \sqrt[r]n \lfloor.

Syntax:

m := IntRoot(r, n)

boolean	m
integer	n
integer	r

IntToChar

Returns the integer with (ASCII) code i, where 0 <= i<256

Syntax:

c := IntToChar(i);

character	c
integer	i

IntValuation

Returns the p-adic valuation of an integer

Syntax:

v := IntValuation(p, n);

integer	v
integer	p
integer	n

IntXGcd

Extended Euclidean algorithm.

Syntax:

G := IntXGcd (a1,a2);
G := IntXGcd (L);

list	G
integer	a1
integer	a2
list	L

IntZeta

Returns the value of the Riemann zeta-function

Syntax:

y := IntZeta(n);

real	y
integer distinct from 1	n

IntegralPoints

Computes all integral points on an elliptic curve in normal form.

Syntax:

L := IntegralPoints(k);
L := IntegralPoints(a,b);

list	L
integer	k
integer	a
integer	b

IsAlff

Returns whether an object is an algebraic function field.

Syntax:

b := IsAlff(F);

boolean	b
object	F

IsAlffDiff

Returns whether argument is an alff differential.

Syntax:

b := IsAlffDiff(a);

boolean	b
alff differential	a

See also: AlffDiff

IsAlffDivisor

Returns true if and only if the parameter is an algebraic function field divisor.

Syntax:

b := IsAlffDivisor(D);

boolean	b
arbitrary object	D

IsAlffElt

Returns true iff the argument is of type "KANT function field order elt".

Syntax:

b := IsAlffElt(T);

boolean	b
function field order element	T

IsAlffPlace

Returns whether an object is an algebraic function field place.

Syntax:

b := IsAlffPlace(P);

boolean	b
arbitrary object	P

IsChar

Returns true if the argument is a character, false otherwise.

Syntax:

b := IsChar(c);

character	c
boolean	b

IsEcc

Returns true iff the argument is an elliptic curve.

Syntax:

b := IsEcc(E);

boolean	b
	E

IsElt

Returns true iff the argument is of type "KANT algebraic element".

Syntax:

b := IsElt(x);

boolean	b
	x

IsFF

Returns whether object is a finite field or not.

Syntax:

b := IsFF(k);

boolean	b
object	k

IsFFElt

Returns true iff the argument is of type "KANT finite field elt".

Syntax:

b := IsFFElt(a);

boolean	b
	a

IsIdeal

Returns true iff the argument is an ideal.

Syntax:

b := IsIdeal(x);

boolean	b
	x

IsInt

Returns true iff the argument is an integer.

Syntax:

b := IsInt(x);

boolean	b
	x

See also: TYPE, IsIdeal, IsPoly, IsOrder, IsElt,

IsLat

Returns true iff the argument is of type "KANT lattice".

Syntax:

b := IsLat(a);

boolean	b
	a

IsMat

Test if an object is or is convertable into a matrix.

Syntax:

bool := IsMat(M);

a matrix-like object	M
a boolean if conversion is possible	bool

IsMat

Returns true iff the argument is of type "KANT matrix".

Syntax:

b := IsMat(x);

boolean	b
	x

IsModule

Returns true iff the argument is a module.

Syntax:

b := IsModule(m);

boolean	b
object	m

IsOrder

Returns true iff the argument is an order.

Syntax:

b := IsOrder(x);

boolean	b
	x

IsPoly

Returns true iff the argument is a polynomial.

Syntax:

b := IsPoly(f);

boolean	b
	f

IsPrime

Returns true iff the argument is either a prime number or a prime ideal.

Syntax:

b := IsPrime(x);

boolean	b
	x

IsQf

Returns whether object is a rational function field or not.

Syntax:

b := IsQf(k);

boolean	b
object	k

IsQp

Checks wether k is an element of a p-adic field.

Syntax:

b := IsQp(k);

p-adic element	k
boolean	b

See also: Qp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpLog, QpPrime, QpSqrt, QpValuation

IsQpElt

Returns true if an element is a p-adic field element.

Syntax:

b := IsQpELt(a);

p-adic element	a
boolean	b

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpLog, QpPrime, QpValuation

IsRecType

Returns true iff the argument is an record with an Type enry containing the correct string.

Syntax:

flag := IsRecType(a, type);

boolean	flag
anything	a
string	type

IsThue

Returns true iff the argument is a Thue object.

Syntax:

b := IsThue(x);

boolean	b
	x

JBessel

Returns the value of the J-Bessel function.

Syntax:

z := JBessel(nu,x);

integer	nu
real	x
real	z

JacobiSymbol

Returns the Jacobi symbol of two integers.

Syntax:

y := JacobiSymbol(n, m);

integer	n
integer	m

KASHLEVEL

Reads or sets different defaults for KASH.

Syntax:

x := KASHLEVEL(s);
x := KASHLEVEL(s,level);

integer	x
string	s
integer	level

KBessel

Returns the value of the K-Bessel function.

Syntax:

z := KBessel(s,x);

complex	s
real	x
complex	z

LOFILES

Lists all open files, for debugging purposes only.

Syntax:

LOFILES();

Lat

Creates a (relative) lattice.

Syntax:

Lambda := Lat(M,["basis"|"gram"]);
Lambda := Lat(o [,"mink"|"unit"]);
Lambda := Lat(a);
Lambda := Lat(Delta,L);
Lambda := Lat(Delta,M [,"trans"|"basis"|"gram"]);
Lambda := Lat(Module);

lattice	Lambda
matrix	M
order	o
ideal	a
list	L
lattice	Delta

LatBasis

Return the basis of a lattice.

Syntax:

M := LatBasis(Lambda);

matrix	M
lattice	Lambda

LatCholesky

Returns a Cholesky type decompostion of the gram matrix of a lattice.

Syntax:

q := LatCholesky(Lambda);

matrix	q
lattice	Lambda

LatDisc

Computes the discriminant of a lattice.

Syntax:

disc := LatDisc(Lambda);

real	disc
lattice	Lambda

LatElt

Creates a lattice element.

Syntax:

a := LatElt(Lambda, L); a := LatElt(Lambda, v);

lattice elt	a
lattice	Lambda
list of reals	L
vector over reals	v

LatEltLength

Computes the length of a lattice element.

Syntax:

r := LatEltLength(a);

real	r
lattice elt	a

LatEltToList

missing shortdoc

Syntax:

L := LatEltToList(a);

list	L
lattice element	a

LatEltVec

Computes the corresponding point of the R^n.

Syntax:

v := LatEltVec(le);

vector	v
lattice element	le

LatEnum

Enumeration of lattice points.

Syntax:

ok := LatEnum(Lambda);

boolean	ok
lattice	Lambda

LatEnumElt

Returns the current element of an enumeration process.

Syntax:

a := LatEnumElt(Lambda);

lattice elt	a
lattice	Lambda

LatEnumLowerBound

Reads or sets the lower bound for enumeration.

Syntax:

lbound := LatEnumLowerBound(Lambda [,lbound]);

real	lbound
lattice	Lambda

LatEnumPrec

Sets the precision used by the enumeration function LatEnum.

Syntax:

s := LatEnumPrec(Lambda);
s := LatEnumPrec(Lambda,"short");
s := LatEnumPrec(Lambda,"long");

list	s
lattice	Lambda

LatEnumRefVec

Reads or sets a reference vector for enumeration.

Syntax:

v := LatEnumRefVec(Lambda [,v]);

vector	v
lattice	Lambda

LatEnumReset

Resets the enumeration environment of a lattice.

Syntax:

LatEnumReset(Lambda);

lattice

  Lambda

LatEnumUpperBound

Reads or sets the upper bound for enumeration.

Syntax:

ubound := LatEnumUpperBound(Lambda [,ubound]);

real	ubound
lattice	Lambda

LatFinckeReduce

Performs a reduction on a lattice specially suited for enumeration.

Syntax:

Lambda2 := LatFinckeReduce(Lambda1);

lattice	Lambda2
lattice	Lambda1

LatGram

Computes the Gram matrix of a lattice.

Syntax:

M := LatGram(Lambda);

matrix	M
lattice	Lambda

LatLLL

Performs a LLL-reduction on a lattice.

Syntax:

Lambda2 := LatLLL(Lambda1);

lattice	Lambda2
lattice	Lambda1

LatShortestElt

Finds shortest elements in a lattice.

Syntax:

L := LatShortestElt(Lambda);
L := LatShortestElt(Lambda,m);
L := LatShortestElt(Lambda,"all");

list of lattice elements	L
lattice	Lambda
integer	m

LatSuccMins

Computes the successive minima of a lattice.

Syntax:

L := LatSuccMins(Lambda);

list	L
lattice	Lambda

Lcm

Returns least common multiple of list of arguments.

Syntax:

lcm := Lcm(L);
lcm := Lcm(a, b);

integer or polynomial or ideal	lcm
list of integers of polynomials or ideals	L
integer or polynomial or ideal	a
integer or polynomial or ideal	b

Li

Returns the value of the logarithmic integral.

Syntax:

a := Li(b);

real	a
real or integer	b

ListApplyListAdd

Computes a linear combination.

Syntax:

a := ListApplyListAdd(L, S);

element	a
lists	L, S	of elements

ListApplyMatAdd

Computes linear combinations.

Syntax:

S := ListApplyMatAdd(L, M);

lists	S, L	of elements
matrix	M	for linear combinations

ListSplit

Splits a list into equal parts. All parts but the last have equal length as specified by the parameter n >= 1.

Syntax:

S := ListSplit(L, n);

list	S	of lists of length <= n
list	L	to be split
integer	n	block length

Log

Returns the principal value of the natural logarithm.

Syntax:

y := Log(x);
y := Log(b,x);

complex or real	y
complex or real or rational or integer	x
complex or real or rational or integer	b

Mat

Creates a matrix.

Syntax:

M := Mat([S,] L);

matrix	M
ring	S
list	L

MatCharPoly

Returns the characteristic polynomial of a matrix.

Syntax:

f := MatCharPoly(M);

matrix	M
Polynomial	f

MatCoef

Returns the coefficient ring of a matrix.

Syntax:

S := MatCoef(M);

ring	S
matrix	M

MatCols

Returns the number of columns of a matrix.

Syntax:

n := MatCols(M);

integer	n
matrix	M

MatDet

Computes the determinant of a square matrix.

Syntax:

d := MatDet(M);d := MatDet(M, bound);

element of the ring the matrix entries come from	d
matrix	M

MatDiag

missing shortdoc

Syntax:

M := MatDiag(S, L);

matrix	M
ring	S
list	L	diagonal entries

MatEchelon

Returns the rank and an upper row echelon form of a matrix together with a transformation matrix. If the echelon form cannot be computed over the ring, it is computed over its field of fractions.

Syntax:

L := MatEchelon(M);

list	L
matrix	M

MatElt

Returns or modifies an matrix entry or row.

Syntax:

a := MatElt(M, row, col);
a := M[row][col];
M[row][col] := a;
r := M[row];
M[row] := L;

element of the ring the matrix entries come from	a
matrix	M
small integer	row
small integer	col
an object of the type "KANT matrow"	r
either a list, a "KANT matrix", or a "KANT matrow"	L

MatHermiteColLower

missing shortdoc

Syntax:

H := MatHermiteColLower(M);

matrix	H
matrix	M

MatHermiteColLowerTrans

Computes the lower column Hermite normal form together with a transformation matrix.

Syntax:

L := MatHermiteColLowerTrans(M);

list	L
matrix	M

MatHermiteColModUpper

Computes the modular upper column Hermite normal form of an integer matrix.

Syntax:

H := MatHermiteColModUpper(k, M);

matrix	H
integer	k
matrix	M

MatHermiteColUpper

missing shortdoc

Syntax:

H := MatHermiteColUpper(M);

matrix	H
matrix	M

MatHermiteColUpperTrans

Computes the upper column Hermite normal form together with a transformation matrix.

Syntax:

L := MatHermiteColUpperTrans(M [,"int"]);

list	L
matrix	M

MatHermiteRowMod

Computes the modular row Hermite normal form of an integer matrix.

Syntax:

H := MatHermiteRowMod(k, M);

matrix	H
integer	k
matrix	M

MatId

Returns the identity matrix of given dimension.

Syntax:

I := MatId(S, n);

matrix	I
ring	S
small integer	n

MatIndex

Computes the determinant of a row normal form of an integral matrix.

Syntax:

n := MatIndex(m);

integer	n	index
matrix	m	over Z

MatInv

Returns the inverse matrix (if possible).

Syntax:

A := MatInv(M);

matrix	A
matrix	M

MatKernel

Returns a basis for the kernel of the given matrix.

Syntax:

K := MatKernel(M);
K := MatKernel(M, d);

matrix	K
matrix	M
integer	d

MatLLL

Performs an LLL reduction on the columns of an integer or real matrix.

Syntax:

L := MatLLL(M [,r]);
L := MatLLL(M,"short" | "long" [,r]);

list of `A`, `T`	L
matrix	M
LLL constant (default 0.75)	r

MatMLLL

Performs an MLLL reduction on the columns of an integer or real matrix.

Syntax:

L := MatLLL(M);

list	L
matrix	M

MatMinPoly

Returns the minimal polynomial of a matrix.

Syntax:

f := MatMinPoly(M);

matrix	M
Polynomial	f

MatMove

missing shortdoc

Syntax:

V := MatMove (M, r);

matrix	V
matrix	M
ring	r

MatRows

Returns the number of rows of a matrix.

Syntax:

n := MatRows(M);

integer	n
matrix	M

MatSmith

missing shortdoc

Syntax:

L := MatSmith(M);

list	L
matrix	M

MatSmithTrans

Computes the Smith normal form together with transformation matrices.

Syntax:

L := MatSmithTrans(M);

list	L
matrix	M

MatSolve

Solves a linear equation.

Syntax:

L := MatSolve (A, b);
L := MatSolve(A, b, N);

list	L
matrix	A
matrix	b
integer	N	a module

MatSym

missing shortdoc

Syntax:

M := MatSym(S, L);

matrix	M
ring	S
list	L

MatSymDiag

missing shortdoc

Syntax:

L := MatSymDiag (A);

list	L
matrix	A

MatToColList

Returns a list of the columns.

Syntax:

L := MatToColList(M);

list	L
matrix	M

MatToRowList

Returns a list of the rows.

Syntax:

L := MatToRowList(M);

list	L
matrix	M

MatTrace

Computes the trace of a matrix.

Syntax:

t := MatTrace(M);

element of the ring the matrix elements come from	t
matrix	M

MatTrans

Returns the tranpose of a matrix.

Syntax:

A := MatTrans(M);

matrix	A
matrix	M

Min

Computes the minimum of an ideal.

Syntax:

n := Min(a);

int \| rational \| ideal	n
ideal	a

MinPoly

Computes the minimal polynomial of an algebraic element, a matrix, or an alff element

Syntax:

p := MinPoly (a [,PA]);
p := MinPoly (a [,O]);
p := MinPoly( M );
p := MinPoly(a);

polynomial	p
polynomial algebra	PA
suborder	O
algebraic element alff order element	a
matrix	M

Module

Creates a module over an order.

Syntax:

M1 := Module([IL,] M);
M1 := Module(EL);

module	M1
list	IL	of ideals over a maximal order O
matrix	M	of algebraic elements over O
list	EL	of relative algebraic elements of o

ModuleDen

Returns the denominator of the given module.

Syntax:

den := ModuleDen(M);

integer	den
module	M

ModuleDet

Returns the determinant of the given module.

Syntax:

det := ModuleDet(M)

module	M
ideal	det

ModuleDual

Computes the dual module of the given module.

Syntax:

M2 := ModuleDual( M1 );

modules

  M1, M2

ModuleId

Returns the trivial module over an order with a certain degree.

Syntax:

m := ModuleId(o, n);

Module	m
order	o
positive integer	n	the degree

ModuleIdeals

Retrieves the coefficient ideals of the module representation.

Syntax:

L := ModuleIdeals(M);

list	L	of ideals
module	M

ModuleIntersection

Computes the intersection of two modules.

Syntax:

M :=ModuleIntersection(M1, M2 [,d | I] [,"PBNF"] [,"lower"] );

modules	M, M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: ModuleConcat, ModuleNF

ModuleIntersectionVS

Computes the intersection of a module and a VS.

Syntax:

M :=ModuleIntersectionVS(M1, mat );

modules	M, M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: ModuleConcat, ModuleNF

ModuleMap

Computes the image and kernel under the multiplication by map

Syntax:

L :=ModuleMap(M1, mat [ , "nf" | "kernel" ] );

modules	M, M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: ModuleConcat, ModuleNF

ModuleMatrix

Retrieves the matrix of the module representation.

Syntax:

m := ModuleMatrix(M);

matrix	m
module	M

See also: Module

ModuleMember

Tests if a column vector is element of the given module.

Syntax:

L:=ModuleMember(M,V|EL);

list	L	see below
module	M
vector	V
list	EL	of n algebraic elements of O

ModuleModul

Returns an ideal which can be used for reduction.

Syntax:

a := ModuleModul(M)

ideal	a
module	M

ModuleMove

Changes the coefficient order of the module.

Syntax:

M2 := ModuleMove(M1, o);

same type as M1	M2
module \| list of modules	M1
order	o

ModuleNF

Computes a normal form of the given module.

Syntax:

M2 := ModuleNF(M1 [,d|I] [,"PBNF"] [,"lower"] );

modules	M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: Module

ModuleOrder

Retrieves the order over which the module is defined.

Syntax:

o := ModuleOrder(M);

order	o
module	M

See also: Module

ModuleSmith

Computes the structure of the quotient module.

Syntax:

L :=ModuleSmith(M1, M2, ["U" | "V" | "UV" );

modules	M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: ModuleConcat, ModuleNF

ModuleSteinitz

Computes a Steinitz form of the given module.

Syntax:

M2 := ModuleSteinitz(M1);

modules

  M1, M2

See also: Module

ModuleUnion

Computes the union of two modules.

Syntax:

M :=ModuleUnion(M1, M2 [,d|I] [,"PBNF"] [,"lower"] );

modules	M, M1, M2
integer	d	used for reduction
ideal	I	used for reduction

See also: ModuleConcat, ModuleNF

Move

Moves an object to another structure (order, ring)

Syntax:

a := Move(L, O);
N := Move(M, r);
a := Move(f, r)

integer or polynomial or ideal	a
list of integers of polynomials or ideals	L
Order	O
matrix	N, M
polynomial	f
ring	r

MultiCounterInc

Increments a multi counter subject to a bound.

Syntax:

b := MultiCounterInc(L, n);

boolean	b
list	L
integer \| list	n

MultiCounterInit

Initializes a multi counter.

Syntax:

L := MultiCounterInit(n);

list	L
integer	n

NextPrime

Returns the smallest rational prime greater a given rational integer.

Syntax:

p := NextPrime(n);

integer	p
integer	n

Nice

If called with no parameter, this function returns the current priority (nice) value. Otherwise the priority (nice) value is set to the parameter.

Syntax:

Nice(v);
v := Nice();

small integer

Norm

Computes the norm of an algebraic number, an ideal, a polynomial or an alff element or divisor.

Syntax:

n := Norm(a);
n := Norm(b, o);

int \| rational \| algebraic number \| polynomial	n
algebraic element \| ideal \| polynomial \| alff element \| ff element	a
algebraic element \| ff element	b
order	o

Num

Returns the numerator of an object.

Syntax:

d := Num( q );
d := Num( a );
d := Num(I);

integer	d
quotient field element or polynomial	d
rational	q
algebraic function field order element	a
algebraic element	a
quotient field element or polynomial	q
ideal	I

Open

(Re)opens a file.

Syntax:

f := Open(name, mode);
ok := Open(f);

File	f
string	name
string	mode
boolean	ok

See also: BagRead, BagWrite, Close, ECHOon, ECHOoff, FLDin, FLDout, LOFILES

Order

Returns the order defined by the given arguments.

Syntax:

o1 := Order (f);
o1 := Order (o,d,alpha);
o1 := Order (o,T,d);
o1 := Order (o,T,L);
o1 := Order (o,L);

order	o1
order	o
polynomial	f
integer	d
matrix	T
algebraic element	alpha
list	L

OrderAbs

Creates an absolute extension from a simple relative extension or creates a simple relative extension from a double relative extension.

Syntax:

Oa := OrderAbs(O);
Oa := OrderAbs(O,"no hom");

order	Oa
order	O

OrderAutomorphisms

Computes or stores automorphisms of the given extension.

Syntax:

aut := OrderAutomorphisms(o);
aut := OrderAutomorphisms(o, L);
aut := OrderAutomorphisms(o, "normal");
aut := OrderAutomorphisms(o, "abel");

list	aut	list of automorphisms
order	o	the given order
list	L	list of some known automorphisms

OrderAutomorphismsAbel

Computes of the given Abelian extension.

Syntax:

aut := OrderAutomorphismsAbel(o);

logical	aut	IsAbelian
order	o	the given order

OrderAutomorphismsNormal

Computes automorphisms of the given normal extension.

Syntax:

aut := OrderAutomorphismsNormal(o);

list	aut	list of automorphisms
order	o	the given order

OrderBach

Returns the value of the floor function of the Bach bound of the algebraic number field which is generated by the given order.

Syntax:

b := OrderBach(O);

integer	b
order	O

OrderBasis

Returns a basis whose elements are elements of the given order with a denominator.

Syntax:

L := OrderBasis(o);
L := OrderBasis(o, O);

list of algebraic elements	L
order	o
order	O

OrderBasisIsPower

Returns true iff the given order is an equation order.

Syntax:

B := OrderBasisIsPower(o);

boolean	B
order	o

OrderBasisIsRel

Returns true iff the basis of an order is given by a transformation matrix.

Syntax:

b := OrderBasisIsRel(o);

boolean	b
order	o

OrderClassGroup

Computes the class group.

Syntax:

L := OrderClassGroup( O [,b] [,"fast"] [,"Euler"] );

list	L
order	O
integer	b

OrderClassGroupCheck

Checks the results of class group computations done by using the Euler product.

Syntax:

OrderClassGroupCheck(o, [[lb,] ub ]  |  [p, "pmax"]);

order	o
integer	ub	upper resp.~ lower bound
integer	p	prime number

OrderClassGroupCyclicFactors

Returns a list with the generators of the cyclic factors.

Syntax:

L := OrderClassGroupCyclicFactors(O);

list	L
order	O

OrderClassGroupCyclicFactorsPrincipal

Returns a list of the generators of the cyclic factors of the class group to the power of their orders.

Syntax:

L := OrderClassGroupCyclicFactorsPrincipal(O, ["raw"]);

list	L
order	O

OrderClassGroupFactorBasisProve

Checks whether a small factorbasis generates the same subgroup of the classgroup as a large factorbasis.

Syntax:

r := OrderClassGroupFactorBasisProve(o, lb, ub);

boolean	r
order	o
integer	lb, ub	lower resp.~upper bound

OrderCoefIdeals

Returns the list of coefficient ideals of a relative order.

Syntax:

L := OrderCoefIdeals(o);

list of ideals	L
relative order	o

OrderCoefOrder

Returns the coefficient ring of a given order.

Syntax:

c := OrderCoefOrder(o);

order, ring of integers	c
order	o

OrderCyclotomic

Computes the n'th cyclotomic field.

Syntax:

o := OrderCyclotomic(n);

order	o
integer	n

OrderCyclotomicRealSubfield

Computes the maximal real subfield of the n'th cyclotomic field.

Syntax:

o := OrderCyclotomicRealSubfield(n);

order	o
integer	n

OrderDeg

Returns the relative degree of the given order.

Syntax:

d := OrderDeg(o);

integer	d
order	o

OrderDegAbs

Returns the absolute degree of the given order.

Syntax:

d := OrderDegAbs(o);

integer	d
order	o

OrderDisc

Returns the discriminant of the given order.

Syntax:

D := OrderDisc(o);

integer or ideal	D
order	o	over Z or over a maximal order

OrderEquationOrder

Returns the equation suborder of the given order.

Syntax:

oe := OrderEquationOrder(o);

order	oe
order	o

OrderExcepSequence

Computes a sequence of exceptional units of maximal length in a given order.

Syntax:

L := OrderExcepSequence (o);

list	L	exceptional sequence
order	o

OrderFincke

Computes or sets the Fincke constants.

Syntax:

OrderFincke(o);
OrderFincke(o, x);
OrderFincke(o, x, y);

order	o
real	x
real	y

OrderGalois

Computation of Galois groups.

Syntax:

OrderIndex

Returns the index of the given order relative to its suborder.

Syntax:

I := OrderIndex(o);

integer	I
order	o

OrderIndexFormEquation

Solves an index form equation.

Syntax:

L := OrderIndexFormEquation(o,index);

list	L
order	o
integer	index

OrderInstallHom

Installs an embedding from the first given order into the second given order.

Syntax:

OrderInstallHom(o1,o2,alpha);
OrderInstallHom(o1,o2,b);

order	o1
order	o2
algebraic element	alpha
boolean	b

OrderIsMaximal

Tests whether the given order is known to be maximal or not.

Syntax:

B := OrderIsMaximal(o);

boolean	B
order	o

OrderIsSubfield

Tests Quo( o|1)\subseteq Quo( o|2).

Syntax:

OrderIsSubfield(o1, o2);

order	o1
order	o2

OrderKextDisc

Computes the relative discriminant of a Kummer extension of prime degree.

Syntax:

disc := OrderKextDisc(F);

ideal	disc
order	F

OrderKextGenAbs

Computes a set of absolute generators for the ring of integers.

Syntax:

Gen := OrderKextGenAbs(F);

list	Gen	of algebraic elements
order	F

OrderKextGenRel

Computes relative generators for the ring of integers.

Syntax:

Gen := OrderKextGenRel(F);

list	Gen	of algebraic elements
order	F

OrderKextModularPower

Computes a maximal exponent.

Syntax:

L := OrderKextModularPower(pI, mu);

list	L	containing alpha and l
ideal	pI	a prime ideal
algebraic element	mu

OrderLLL

Creates an order with LLL-reduced basis.

Syntax:

o1 := OrderLLL(O);

order	o1
order	o

OrderMaximal

Returns the maximal overorder of an order.

Syntax:

O := OrderMaximal( def);
O := OrderMaximal( def, str);

order	O
see below	def
up to 4 optional strings	str

OrderMerge

Computes extension fields which contain the given ones.

Syntax:

L := OrderMerge(o1,o2);

list	L	return value
order	o1
order	o2

OrderMinIdeal

Computes a non-zero integral ideal of smallest norm.

Syntax:

a := OrderMinIdeal (o);

ideal	a
order	o

OrderMinkowski

Returns the floor of the Minkowski bound of the algebraic number field which is generated by the given order.

Syntax:

m := OrderMinkowski(O);

integer	m
order	O

OrderNormEquation

Solves a (relative) norm equation.

Syntax:

L := OrderNormEquation(o, a [,n | "all" [,"exact" | "abs" | "ineq"]]);

list	L
int \| element of the coefficient ring	a
int	n

OrderPMaximal

Computes the p-maximal overorder of an order.

Syntax:

op := OrderPMaximal(o,p, str);
op := OrderPMaximal(o,p,b, str);

order	op
order	o
rational prime or prime ideal	p
integer	b
up to 3 optional strings	str

OrderPoly

Returns the defining polynomial of the associated equation order

Syntax:

f := OrderPoly(P,o);
f := OrderPoly(o);

polynomial	f
polynomial algebra	P
order \| algebraic function field order	o

OrderPolyHenselLift

Calculates a p-adic factorization of the defining polynomial of an order.

Syntax:

L := OrderPolyHenselLift(o, d, p, k);

list	L
order	o
integer	d
prime	p
integer	k

OrderPrec

Sets or returns the internal precision for calculations in orders.

Syntax:

P := OrderPrec(p);
P := OrderPrec();
L := OrderPrec(o,p);
L := OrderPrec(o,p,u);
L := OrderPrec(o);

small integer	P
small integer	p
order	o
small integer	u
list	L	precisions of both real rings belonging to `o`

OrderPrintFlags

Set or get Flags to reduce or extend the output level of print_order.

Syntax:

OrderPrintFlags(a);
a := OrderPrintFlags();

record

OrderReg

Returns the regulator of the current maximal system of independent units.

Syntax:

x := OrderReg(o);
x := OrderReg(o,"classgroup");
x := OrderReg(o, reg);

real	x
order	o
real	reg

OrderRegLowBound

Returns a lower bound for the regulator of the given order.

Syntax:

x := OrderRegLowBound(o);
x := OrderRegLowBound(o, reg);

real	x
order	o
real	reg

OrderRelNormEq

Calculates an element of given norm.

Syntax:

elt := OrderRelNormEq(O,n [, "true"]);

order	O
list	elt	list of elements in O
element	n	element in coefficient ring of O

OrderRelUnits

Computes the relative units.

Syntax:

L := OrderRelUnits(o [,S  |  I] [,"raw"]);

list	L	of L1, T, hs or L2, hs
list	L1	of algebraic integers
list	L2	of S-units
matrix	T	transformation matrix
integer	hs	S-class number
order	o
list	S	of pairwise distinct prime ideals.
ideal	I

OrderRelativeOrder

Computes a relative representation of the given orders.

Syntax:

Or := OrderRelativeOrder(O, o);

order	Or
order	O
order	o

OrderSUnits

Computes the S-unit group.

Syntax:

L := OrderSUnits(o [,S  |  I] [,"raw"]);

list	L	of L1, T, hs or L2, hs
list	L1	of algebraic integers
list	L2	of S-units
matrix	T	transformation matrix
integer	hs	S-class number
order	o
list	S	of pairwise distinct prime ideals.
ideal	I

OrderSUnitsPositive

Returns a list of positive S-units

Syntax:

P := OrderSUnitsPositive(L);

list	P
list	L

OrderSetMaximal

Sets the OrderIsMaximal flag in the given order.

Syntax:

OrderSetMaximal(o);
OrderSetMaximal(o,flag);

order	o
optional boolean	flag

OrderSetTorsionUnit

Sets a generator of the torsion unit group in the given order.

Syntax:

OrderSetTorsionUnit(o,alpha,r);

order	o
algebraic element	alpha
integer	r	rank

OrderShort

Tries to find a "better" primitive polynomial for the given order.

Syntax:

o1 := OrderShort(o);
o1 := OrderShort(o, modus);
o1 := OrderShort(o, modus, iterations);

order	o1
order	o
integer	modus
integer	iterations

OrderShortAbs

Creates an absolute extension from a simple relative extension and tries to find a "good" primitive polynomial for the given order.

Syntax:

Oa := OrderShortAbs(O);

order	Oa
order	O

OrderSig

Returns the signature of the number field defined by the given order.

Syntax:

L := OrderSig(o);

list	L
order	o

OrderSimplify

Returns the given order in a simplified representation.

Syntax:

o1 := OrderSimplify(o);

order	o1
order	o

OrderSplittingField

Computes the normal closure of an algebraic number field.

Syntax:

O := OrderSplittingField (o);

order	O
order	o

OrderSubOrder

Returns a suborder of the given order.

Syntax:

os := OrderSubOrder(o);

order or boolean	os
order	o

OrderSubfield

Returns all non-trivial subfields of given degree m. If no m is specified, all subfields are calculated.

Syntax:

L := OrderSubfield(o);
L := OrderSubfield(o, m);

list	L	list of suborders
order	O	the given order
small integer	m	the prescribed degree of subfields

OrderSubfieldSub

Calculates subfields with specified block systems.

Syntax:

L := OrderSubfieldSub(o, p);
L := OrderSubfieldSub(o, p, d);
L := OrderSubfieldSub(o, p, d, L1);

list	L
order	o
integer	p
integer	d
list	L1

OrderTorsionUnit

Returns a generator for the group of torsion units of the given order.

Syntax:

u := OrderTorsionUnit(o);

algebraic element	u
order	o

OrderTorsionUnitRank

Returns the number of roots of unity in the given order.

Syntax:

r := OrderTorsionUnitRank(o);

integer	r	number of roots of unity
order	o

OrderTraceMat

Returns the trace matrix of the order.

Syntax:

M := OrderTraceMat(o);

Matrix	M
order	o

OrderTransformationMatrix

Returns a list containing information about the transformation from a suborder of the given order to the given order.

Syntax:

L := OrderTransformationMatrix(O);

list	L
order	O

See also: OrderBasis, OrderCoefIdeals, Order

OrderUnitsAreFund

Tests whether the current unit system of the given order is known to be fundamental or not.

Syntax:

b := OrderUnitsAreFund(o);
b := OrderUnitsAreFund(o,c);

boolean	b, c
order	o

OrderUnitsEquation

Solves a unit equation.

Syntax:

L := OrderUnitsEquation (alpha,beta,gamma);
L := OrderUnitsEquation (alpha,beta);

list	L
algebraic elements	alpha, beta, gamma

OrderUnitsExcep

Computes all exceptional units of an order.

Syntax:

L := OrderUnitsExcep(o);
L := OrderUnitsExcep(o,"orbits"|"list");
n := OrderUnitsExcep(o,"number");

list	L
integer	n
order	o

OrderUnitsFund

Returns a list whose entries are a maximal set of fundamental units (algebraic numbers) of the given order.

Syntax:

L := OrderUnitsFund(o);

list	L
order	o

OrderUnitsIndep

Computes a maximal system of independent units.

Syntax:

L := OrderUnitsIndep(o);
L := OrderUnitsIndep(o,"classgroup");
L := OrderUnitsIndep(o,"classgroup","list");

list	L
order	o

OrderUnitsLLL

Computes a LLL reduced system of independent units.

Syntax:

OrderUnitsLLL (O);

order

OrderUnitsMerge

Extends a system of units of the given order.

Syntax:

B := OrderUnitsMerge(o,eta);
B := OrderUnitsMerge(o,eta,"append");

boolean	B
order	o
algebraic element	eta	a unit in o

OrderUnitsPFund

Computes the p maximal overgroup of the units in the given order.

Syntax:

OrderUnitsPFund (O,p);

order	O
rational prime	p

PRINTLEVEL

Reads or sets the printlevel for a certain function.

Syntax:

x := PRINTLEVEL(s);
x := PRINTLEVEL(s,level);
x := PRINTLEVEL("all",level);

small integer	x
string	s
small integer	level

Poly

Creates a polynomial.

Syntax:

f := Poly(A, L);

polynomial	f
polynomial algebra	A
list	L

See also: PolyAlg

PolyAlg

Creates a univariate polynomial algebra over a ring or returns the polynomial algebra of a polynomial.

Syntax:

Sx := PolyAlg(S [, name]);

polynomial algebra	Sx
ring or polynomial	S
string	name

PolyAlgCoef

Returns the coefficient ring of a polynomial algebra.

Syntax:

S := PolyAlgCoef(Sx);

ring	S
polynomial algebra	Sx

See also: PolyAlg

PolyDeg

Returns the degree of a given polynomial.

Syntax:

n := PolyDeg(f);

integer	n
polynomial	f

PolyDeriv

Returns the derivative of a given polynomial.

Syntax:

h := PolyDeriv(f);

polynomial	h
polynomial	f

PolyDisc

Returns the discriminant of a polynomial.

Syntax:

d := PolyDisc(f);

discriminant (integer or finite field element)	d
polynomial	f

PolyFactor

Returns the factorization of the given polynomial.

Syntax:

F := PolyFactor(f);
F := PolyFactor(f, p);
F := PolyFactor(f, p, m);

list	F
polynomial	f
prime number	p
integer	m

See also: Factor

PolyGcd

Returns the gcd of two polynomials.

Syntax:

g := PolyGcd(f, h);

polynomial	g
polynomial	f
polynomial	h

PolyHenselLift

Lifts a factorization of a polynomial modulo a prime ideal to a factorization modulo the prime ideal to a given exponent.

Syntax:

PolyHenselLift(f, A, n);

polynomial over an order	f
ideal of the same order	A
integer	n

PolyIsIrreducible

missing shortdoc

Syntax:

b := PolyIsIrreducible(f);

boolean	b
polynomial	f

PolyIsSquarefree

missing shortdoc

Syntax:

b := PolyIsSquarefree(f);

boolean	b
polynomial	f

PolyIsZero

Returns true iff the argument is a polynomial and equal to the zero polynomial

Syntax:

b := PolyIsZero(f);

boolean	b
	f

PolyMakeMonicInOrder

Make f(x) monic by rational transformation, so that the resulting monic polynomial g(x) is defined over o and creates the same algebra as f(x). g(x) may be the same as f(x).

Syntax:

g:= PolyMakeMonicInOrder(f, o);

order	o
polynomial	f
polynomial	g

PolyMakeMonicInZ

Given f(x) in Q[x], the function returns g(x) monic in Z[x], such that g(x) generates the same algebra as f(x).

Syntax:

g:= PolyMakeMonicInZ(f);

polynomial	f
polynomial	g

PolyMove

Tries to compute a representation of a polynomial in a different polynomial algebra.

Syntax:

g := PolyMove (f, S);

polynomial	g
polynomial	f
ring	S

See also: PolyAlg

PolyMoveIntegral

Returns the polynomial moved to integral coefficients.

Syntax:

g := PolyMoveIntegral(f);

polynomial	f
polynomial	g

PolyNewtonLift

Lifts an algebraic element with the Newton lifting method.

Syntax:

beta := PolyNewtonLift(f, alpha, k);
beta := PolyNewtonLift(f, alpha, k, a);

algebraic element	alpha
integer (0 if omitted)	a
polynomial	f
integer	k
polynomial	beta

PolyNorm

Returns the norm of a polynomial.

Syntax:

n := PolyNorm(f);

norm (polynomial)	n
polynomial	f

PolyPowerMod

missing shortdoc

Syntax:

h := PolyPowerMod(f, n, g);

polynomial	g
polynomial	f
polynomial	h
positive integer	n

PolyPrimeList

Returns a list containing all monic prime polynomials of degree d in k[x] for a finite field k.

Syntax:

L := PolyPrimeList(kx, d);

list	L
polynomial algebra	kx	over finite field k
integer	d

PolyPrimeNum

Return the number of monic prime polynomials over a finite field.

Syntax:

f := PolyPrimeNum(kx, d);

polynomial	f
polynomial algebra over k	kx
integer	d

PolyPrimeRandom

Return a random prime polynomial over a finite field.

Syntax:

f := PolyPrimeRandom(kx, d);

polynomial	f
polynomial algebra over k	kx
integer	d

PolyQuotRem

missing shortdoc

Syntax:

L := PolyGcd(f, g);

polynomial	g
polynomial	f
list	L

PolyRedDisc

Returns the reduced discriminant of a separable polynomial.

Syntax:

d := PolyRedDisc(f);

integer	d
polynomial	f

PolyResultant

Computes the resultant of the two given polynomials.

Syntax:

r := PolyResultant (f, g);

polynomial	r
polynomial	f
polynomial	g

PolyRoundFour

Returns the factorization of the given polynomial over the p-adic integers and certificates for the irreducibility of the factors.

Syntax:

L := PolyRoundFour(f);
L := PolyRoundFour(f, p);
L := PolyRoundFour(f, p, m);

list	L
polynomial	f
prime number	p
integer	m

See also: Factor, PolyFactor, OrderMaximal

PolySig

Computes the signature of a monic squarefree polynomial.

Syntax:

s := PolySig(f);

polynomial	f
list	s

PolySwapVars

Changes the variables of the given bivariate polynomial.

Syntax:

g := PolySwapVars (f);

polynomial	g
polynomial	f

PolyToList

Returns the coefficients of a polynomial in a list.

Syntax:

L := PolyToList(f);

list	L
polynomial	f

PolyXGcd

Returns the extended gcd of two polynomials.

Syntax:

l := PolyXGcd(f, h);

list of polynomials	l
polynomial	f
polynomial	h

PolyZeros

Computes roots of a polynomial.

Syntax:

L := PolyZeros (f [[, "complex"|"int"] [, p [, m ]]]);

list	L
polynomial	f

Prec

Sets and gets the precision for real and complex computations in the shell.

Syntax:

Prec(n);
Prec();

integer

PvmClear

Remove all broadcast-jobs, all failed-jobs and all slave-jobs.

Syntax:

PvmClear();

PvmExit

Stops kash-Pvm.

Syntax:

PvmExit

PvmGet

Receives data from PVM.

Syntax:

L := PvmGet();

List

PvmGetAnswer

Waits for a valid answer to arrive from the slave.

Syntax:

L := PvmGetAnswer(true | false);

list

first entry is in {1, 2, 4}, all others arbitrary

See also: PvmGet

PvmGetB

Waits for any data to arrive.

Syntax:

L := PvmGetB();

list

See also: PvmGet

PvmGetEval

Syntax:

 not intended to be called by a user

PvmInit

Starts kash-Pvm on the master.

Syntax:

PvmInit();

PvmKashIsSlave

true iff we are running as slave, false otherwise.

Syntax:

s := PvmKashIsSlave();

boolean

PvmLengthOfQueue

If called with no parameter, this function returns the maximal number of jobs that can be stored in queue. Otherwise the number is set to the parameter.

Syntax:

PvmLengthOfQueue(n1);
n2 := PvmLengthOfQueue();

integer

  n1, n2

PvmMaxRestartSlave

If called with no parameter, this function returns the maximal number of times a slave may be restarted by security system. Otherwise the number is set to the parameter.

Syntax:

PvmMaxRestartSlave(n1);
n2 := PvmMaxRestartSlave();

integer

  n1, n2

PvmMaxRetransmitJob

If called with no parameter, this function returns the maximal number of times a job may be retransmitted by security system. Otherwise the number is set to the parameter.

Syntax:

PvmMaxRetransmitJob(n1);
n2 := PvmMaxRetransmitJob();

integer

  n1, n2

PvmPread

Reads and prints all arrived data in plain style, i.e. as provided by kash.

Syntax:

PvmPread();

See also: PvmGet

PvmRead

Reads (and prints) all data already received from the slave(s).

Syntax:

PvmRead();

See also: PvmGet

PvmSecurity

If called with no parameter, this function returns the current status of the security system (activated or not). Otherwise it is set to the parameter.

Syntax:

PvmSecurity(true|false);
b := PvmSecurity();

boolean

PvmSendAll

Sends data to all slaves.

Syntax:

PvmSendAll(data, data, ...);

PvmSendLast

Sends all parameters to the next free slave.

Syntax:

ok := PvmSendLast(data, data, ...);

PvmSendNext

Sends all parameters to the next free slave.

Syntax:

PvmSendNext(data, data, ...);

PvmSetPrintLevel

Set the pvm_master->print_level to lev. Only for debuging.

Syntax:

PvmSetPrintLevel(lev);

small integer

lev

PvmShowBroadcastJobs

Returns the formatstrings of the broadcast--job as a list. Intendend maily for debugging.

Syntax:

PvmShowBroadcastJobs();

PvmSlaveError

Slave only: Sends all data given as parameter to the master.

Syntax:

KashPvmError(data, data, ...);

See also: PvmGet

PvmSlaveInfo

Returns information of all Slaves in a list of records.

Syntax:

PvmSlaveInfo();

PvmSlavePrint

Slave only: Sends all data given as parameter to the master.

Syntax:

PvmSlavePrint(data, data, ...);

See also: PvmGet

PvmSlaveSend

Slave only: Sends all data given as parameter to the master.

Syntax:

PvmSlaveSend(data, data, ...);

See also: PvmGet

PvmStartSlave

Starts the slave(s).

Syntax:

noSlaves := PvmStartSlave(num);
noSlaves := PvmStartSlave(host);

integer	noSlaves	the number of started slaves
integer	num	the number of slaves to start. Use 0 to start as many as possible.
string	host	start one slave at host.
list	L	List of hostnames to start one slave at.

PvmStopSlave

Kills the slaves. If no argument is given, all slaves will be killed.

Syntax:

noSlaves := PvmStopSlave();
noSlaves := PvmStopSlave(stid);
noSlaves := PvmStopSlave(L);

integer	noSlaves	the number of killed slaves
integer	stid	slave-tid to kill
list	L	List of slave-tids to kill

PvmStoreOrders

Toggels/ Queries some internal flags.

Syntax:

PvmStoreOrders

PvmUse

Enables or disables the use of KANT-PVM. Initial state is false. If no argument is given, this function gets the status of PvmUse.

Syntax:

PvmUse(true|false);
s := PvmUse();

boolean

PvmUseMastersHost

Enables or disables the use of the master's host for PvmStartSlave. Initial state is false. If no argument is given, it gets the status of PvmUseMastersHost.

Syntax:

PvmUseMastersHost(true|false);
s := PvmUseMastersHost();

boolean

PvmUseMsg

Enables or disables output for debugging.

Syntax:

PvmUseMsg(true|false);
s := PvmUseMsg();

boolean

PvmUseWatch

Enables or disables the PVM-Watch for debugging. If h is given, the PVM-Watch starts on this host. Initial state is false. If no argument is given, it gets the status of PvmUseWatch.

Syntax:

PvmUseWatch(x, h);
s := PvmUseWatch(x);
s := PvmUseWatch();

boolean	x
string	h	hostname
boolean	s

Q

Predefined constant: Rational field Q.

Syntax:

Q;

ring

QfName

missing shortdoc

Syntax:

s := QfName(S);

string	s
polynomial algebra or quotient field	S

QfRank

missing shortdoc

Syntax:

n := QfRank(S);

integer	n
polynomial algebra or quotient field	S

QfScalarRing

missing shortdoc

Syntax:

A := QfScalarRing(S);

ring	A
polynomial algebra or quotient field	S

QfeDen

This function returns the denominator of a rational function (quotient field element). Works also for polynomials.

Syntax:

d := QfeDen(f);

quotient field elements

d,f

QfeDeriv

Computes the derivation of an rational function.

Syntax:

dhdT := QfeDeriv(h);

qf elements

  dhdT, h

QfeNum

This function returns the numerator of a rational function (quotient field element). Works also for polynomials.

Syntax:

d := QfeNum(f);

quotient field elements

d,f

QfePthRoot

Returns the p-th root of a rational function with scalars in a finite field of characteristic p, if existent.

Syntax:

r := QfePthRoot(a);

qf element or bool	r	r p-th root of a or `false`
qf element	a

QfeQf

Returns the quotient field in which a rational function (quotient field element) is defined.

Syntax:

F := QfeQf(a);

quotient field	F
quotient field element	a

QfeVal

Computing valuations in a rational function field.

Syntax:

v := QfeVal(p, a);

integer	v
qf elements	p, a

Qp

Creates the p-adic field Q_p

Syntax:

F := Qp(p[,n]);

p-adic field	F
prime number	p
integer	n

See also: IsQp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpLog, QpPrime, QpSqrt, QpValuation

QpElt

Returns an element of Q_p.

Syntax:

k := QpElt(F, n);

p-adic field	F
integer or rational	n
p-adic element	k

See also: IsQp, Qp, QpEltToQ, QpEltQp, QpExp, QpLog, QpPrec, QpPrime, QpSqrt, QpValuation

QpEltQp

This function creates the p-adic field of the element k.

Syntax:

F := QpEltQp(k);

p-adic field element	k
p-adic field	F

See also: IsQp, Qp, QpElt, QpEltToQ, QpPrec, QpExp, QpLog, QpPrime, QpSqrt, QpValuation

QpEltToQ

Computes the rational number q of an element in Q_p.

Syntax:

q := QpEltToQ(k);

p-adic field	F
p-adic element	k
rational number	q

See also: IsQp, Qp, QpElt, QpEltQp, QpExp, QpLog, QpPrec, QpPrime, QpSqrt, QpValuation

QpExp

Computes the exponent function of an element in Q_p.

Syntax:

exp:= QpExp(k);

p-adic element	k
p-adic element	exp

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpLog, QpPrec, QpPrime, QpSqrt, QpValuation

QpLog

Computes the logarithm of an element in Q_p.

Syntax:

l := QpLog(k);

p-adic element	k
p-adic element	l

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpPrime, QpSqrt, QpValuation

QpPrec

Prints or sets the precision of a p-adic field.

Syntax:

k:= QpPrec(F [,n]);

p-adic field	F
integer	n
integer	k

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpExp, QpLog, QpPrime, QpSqrt, QpValuation

QpPrime

This function creates the prime number which defines the p-adic field.

Syntax:

p := QpPrime(F);

p-adic field	F
prime number	p

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpLog, QpSqrt, QpValuation

QpSqrt

Computes the square root of an element in Q_p.

Syntax:

s := QpSqrt(k);

p-adic element	k
p-adic element	s

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpPrec, QpExp, QpLog, QpPrime, QpValuation

QpValuation

Computes the valuation of an element in Q_p.

Syntax:

v := QpValuation(F, k);

p-adic field	F
p-adic element	k
integer	v

See also: IsQp, Qp, QpElt, QpEltToQ, QpEltQp, QpExp, QpLog, QpPrec, QpPrime, QpSqrt

QuotientField

missing shortdoc

Syntax:

QF := QuotientField(S);

quotient field	QF
ring	S

See also: PolyAlg

R

Predefined constant: Real field R.

Syntax:

R;

ring

RandomEcc

Returns a random elliptic curve in Weierstrass form.

Syntax:

E := RandomEcc(F);

elliptic curve	E
finite field	F

RandomElt

Computes a "random" element of a given order.

Syntax:

alpha := RandomElt(R [, l | b, deg | degl]);

element	alpha	of `o`
ring	R	may be an order, an ideal, Z, a module, a polynomial algebra or an function field order.
list	l	of integers
integer	b	equivalent to l := [-b..b]
integer	deg	degree of a random element of the
	polynomial algebra R
list	degl	of positive integers

RandomIdeal

A random ideal over a given order is computed.

Syntax:

id := RandomIdeal(o);

order	o
ideal	id

RandomMatrix

Using RandomMatrix a square matrix with "random" entries is returned.

Syntax:

m := RandomMatrix(R, n, l);

matrix	m	in `R`^{ `n` \times `n`}
ring	R	may be an order, an ideal, Z, a module or an function field order.
integer	n
list	l	of possible (coeffiecents of the) entries.

RandomOrder

Using RandomOrder a random order is computed.

Syntax:

o := RandomOrder(R, n [, l]);

order	o
ring	R	may be Z, an order or an functions field order.
integer	n	degree
list	l

RandomPoly

Produces a monic random polynomial of given degree.

Syntax:

f := RandomPoly(R, d [, l]);

polynomial	f	in `R`[x]
ring	R	may be an order, an ideal, Z, a module or an function field order.
integer	d	degree of f
list	l	of possible (coeffiecents of the) coefficients.

RationalReconstruct

Lifts an integer modulo m to a rational.

Syntax:

q := RationalReconstruct (u,m);

rational	q
integer	u
integer	m

RayCantoneseRemainder

missing shortdoc

Syntax:

elt := RayCantoneseRemainder(m0,minf,elt0,sig);

ideal	m0
list of integers/infinite primes	minf
algebraic number	elt0
list	sig
algebraic number	elt

RayClassField

Given a congruence module, this function computes generating polynomials for the class fields belonging to the cyclic factors of the corresponding ray class group.

Syntax:

c := RayClassField( o | a [, [inf | inf, deg | mat] | deg | mat ] );

list	c	containg polynomials of `o[x]`
order	o	for Hilbert class field computations
ideal	a	used as (part of) an congruence module as for `RayClassGroup`
list	inf	of infinite places as for `RayClassGroup`
integer	deg	if your interesed only in the `deg` part of the class field
matrix	mat	rows of `mat` define a factorgroup of the RayClassGroup

RayClassFieldAbelianTest

Tests if a class field is also abelian over Q.

Syntax:

a:=RayClassFieldIsAbelian(G [,m]);

AbelianGroup	G	defining a class group
boolean	a
integer	m

RayClassFieldArtin

Given an ideal, this function computes the corresponding automorphism.

Syntax:

aut := RayClassFieldArtin(id, O);

an automorphism	aut
ideal	id	coprime to the defining module
order	O	output of `RayClassFieldAuto`

RayClassFieldAuto

Given the output of RayClassField, this function computes a primitive element for the Ray Class Field togther with the automorphisms.

Syntax:

L := RayClassFieldAuto(c);

list	L	containing `O` and a list of automorphisms
list	c	output of `RayClassField`

RayClassFieldIsAbelian

Tests if a class field is also abelian over Q.

Syntax:

a:=RayClassFieldIsAbelian(G [,m]);

AbelianGroup	G	defining a class group
boolean	a
integer	m

RayClassFieldIsCentral

Tests whether the Ray Class Field defined by the given data will be a central extension.

Syntax:

flag := RayClassFieldIsCentral(G [, l]);

boolean	flag
AbelianGroup	G	a quotient from RayClassGroupToAbelianGroup
list	l	of automorphisms, if not present all known automorphisms are used.

RayClassFieldIsNormal

Tests whether the Ray Class Field defined by the given data will be a normal extension.

Syntax:

flag := RayClassFieldIsNormal(G [, l]);

boolean	flag
AbelianGroup	G	a quotient from RayClassGroupToAbelianGroup
list	l	of automorphisms, if not present all known automorphisms are used.

RayClassFieldSplittingField

Computes data to get the splitting field of the Ray Class Field.

Syntax:

sg := RayClassFieldSplittingField(G, l);

AbelianGroup	sg
AbelianGroup	G	quotient of RayClassGroupToAbelianGroup
list	l	of automorphisms. Must be complete, not just generators.

RayClassGroup

Returns the ray class number and the orders of the generators of the ray class group modulo a congruence module.

Syntax:

L := RayClassGroup(m0 [,minf]);
L := RayClassGroup(o [,minf]);

list	L
ideal	m0
list	minf	of integers/infinite primes
order	o

RayClassGroupCyclicFactors

Returns the generators and the orders of the generators of the ray class group modulo a congruence module as computed using RayClassGroup.

Syntax:

L := RayClassGroupCyclicFactors(m0 [,minf]);

list	L
ideal	m0
list	minf	of integers/infinite primes

RayClassGroupToAbelianGroup

Returns the ray class group for a congruence module.

Syntax:

g := RayClassGroupToAbelianGroup(m0 [, minf] [, rels | expo]);

group	g
ideal	m0
list of integers	minf	infinite primes
matrix of integers	rels	gives additional relations for a quotient
integer	expo	equivalent to `rels` = `expo*MatId`.

RayConductor

Calculates the conductor of the ray class group modulo a congruence module.

Syntax:

L := RayConductor(m0 [, minf] [, rels]);

list	L
ideal	m0
list	minf	of integers/infinite primes
matrix	rels	relation matrix over Z

RayConductorTest

Tests iff the given module is the true conductor.

Syntax:

b := RayConductorTest(m0 [, minf] [, rels]);

boolean	b
ideal	m0
list	minf	of integers/infinite primes
matrix	rels	relation matrix over Z

RayDiscSig

Returns the relative discriminant and the absolute signature of the ray class field belonging to a congruence module.

Syntax:

L := RayDiscSig(m0 [,minf] [,rels]);

ideal	m0
list	minf	of integer/infinite primes
matrix	rels	relation matrix over Z
list	L

RayResidueRing

This function computes the multiplicative group of the residue class ring.

Syntax:

L := RayResidueRing(m0 [,minf]);

list	L
ideal	m0
list	minf	of integers/infinite primes

RayResidueRingCyclicFactors

Returns generators (and their orders) for the multiplicative group of the residue class ring of an ideal or a congruence module, as computed by RayResidueRing.

Syntax:

L := RayResidueRingCyclicFactors(m0 [,minf]);

list	L
ideal	m0
list	minf	of integers/infinite primes

RayResidueRingRepToElt

Returns a canonical representative of an element of the multiplicative group of the residue class ring of a congruence module.

Syntax:

b := RayResidueRingRepToElt( r, m0, minf);

algebraic element	b
matrix	r
ideal	m0
list	minf	of integers/infinite primes

RayResidueRingToAbelianGroup

Returns the ray residue ring modulo a congruence module.

Syntax:

g := RayResidueRingToAbelianGroup(m0 [, minf]);

group	g
ideal	m0
list of integers	minf	infinite primes

Re

Returns the real part of a complex number.

Syntax:

a := Re(z);

real	a
complex	z

Read

Read a file containing KASHcommands.

Syntax:

Read(name);

string

  name

ReadLib

The same as Read, but here the file should be in the KASHlib directory and it must have the extension ".g".

Syntax:

ReadLib(name);

string

  name

Round

Returns the integer closest to a given number. Note that if the decimal part of the number given is .5, then the number is rounded up. If the number is negative, it is rounded down.

Syntax:

y := Round(x);

integer	y
real	x

SPrint

Creates a string instead of printing on the screen.

Syntax:

s := SPrint( obj1, obj2, ... );

string	s
string or kash object	obj1, obj2

SScan

Reads kash objects out of a string.

Syntax:

L := SScan( s, fmt, R1, R2, ...);

list	L	containing the kash objects
string	s
string	fmt	format string
rings	R1, R2

SimplexElt

Returns the current point.

Syntax:

L := SimplexElt(s);

list of coordinates of one integral point	L
simplex	s

SimplexInit

Initializes the enumeration of all integral points contained in a bounded simplex.

Syntax:

s := SimplexInit(A, b [ , delta ]);

Simplex	s
real matrix	A	A \in R^{m \times n}
real matrix	b	b \in R^{m}
real	delta	should be 1+\epsilon

SimplexNext

Tries to find a "next" integral point in the given simplex.

Syntax:

ok := SimplexNext(s);

boolean	ok
simplex	s

SimplexReInit

Reinitializes a given simplex. May be used to set delta.

Syntax:

SimplexReInit(s [, delta ]);

simplex	s
real	delta

See also: SimplexNext, SimplexElt, SimplexInit

Sin

Returns the sine of a number.

Syntax:

y := Sin(x);

complex	y
complex	x

Sleep

Lets kash sleep for nSec seconds. Useful for KashPvm

Syntax:

Sleep(nSec);

small integer

  nSec

Solve

Solves an equation or computes the roots of a polynomial.

Syntax:

L := Solve(t, A);
L := Solve(o, a);
L := Solve(f);

list	L
Thue object	t
int	A
order	o
int \| algebraic element	a
polynomial	f

Sqrt

Returns a square root of a number.

Syntax:

y := Sqrt(x);

complex or real	y
complex or real or rational or integer	x

SubfieldAdd

Adds a subfield to the given order.

Syntax:

SubfieldAdd(o, sub, alpha);

order	o
order	sub	subfield
algebraic element	alpha	primitive element of sub in o

SubfieldGet

Returns all computed subfields.

Syntax:

L:=SubfieldGet(o);

list of subfields	L
order	o

SubfieldSetDegreeMax

Sets the maximal number of subfield of degree d.

Syntax:

SubfieldSetDegreeMax(O, d)

order	O
integer	d

Tan

Returns the tangent of a number.

Syntax:

y := Tan(x);

complex	y
complex	x

Thue

Creates a thue object for solving Thue equations.

Syntax:

t := Thue(o);
t := Thue(L);
t := Thue(f);

Thue object	t
order	o
list	L
polynomial	f

ThueEval

missing shortdoc

Syntax:

a := ThueEval(t,x,y);

int	a
Thue object	t
int	x
int	y

ThueSolve

Solves a Thue equation.

Syntax:

L := ThueSolve(t,a [,"exact"|"abs"]);

list	L
Thue object	t
int	a

Time

Toggles the time display.

Syntax:

b := Time([true|false]);

boolean

current status of time display

Trace

Computes the trace of an algebraic number or alff element or a matrix

Syntax:

n := Trace(a);
n := Trace(b, o);

algebraic element \| alff elt \| matrix \| ff element	a
algebraic element \| ff element	b
order	o

TrialDivision

Syntax:

F := TrialDivision(d, b);

list	F
integer	d
integer	b

Trunc

Returns the integer part of a number.

Syntax:

y := Trunc(x);

integer	y
real	x

Valuation

missing shortdoc

Syntax:

d := Valuation( [P,] a);
SHOTDOC Computes the valuation of the argument at a prime (if possible).

integer	d
ideal \| integer \| alff order ideal	P	must be prime.
ideal \| algebraic element \| integer	a	to valuate.
alff order ideal	a	to valuate.

Vec

Returns a vector ( matrix with one column).

Syntax:

v := Vec(r,L);

ring	r
list	L
matrix	M

VecDotProduct

Computes the dot product of two vectors.

Syntax:

r := VecDotProduct (u,v);

list	r
matrix	u
matrix	v

WeierstrassP

Calculates the value of the Weierstrass \wp-function related to the given lattice.

Syntax:

u := WeierstrassP(z, w1, w2);

complex	u
complex	z	a non-zero element of the complex torus C/Zw_1oplusZw_2
complex	w1,w2	complex values with Im(w_1/w_2)>0

World

Returns some information on objects.

Syntax:

World (arg1, arg2, ...);

object 1	arg1
\vdots	\vdots
object n	argn

Z

Predefined constant: Integer ring Z.

Syntax:

Z;

ring

ZIdealCreate

Returns the ideal in Z generated by the given integer.

Syntax:

I := IdealRayClassRep(a);

ideal in Z	I
integer	a

Zx

Predefined constant: polynomial algebra over integer ring.

Syntax:

Zx;

e

Predefined constant: Euler's constant in the current precision of the real field.

Syntax:

e;

real

undefined_function_name

missing shortdoc

Syntax:

undefined_function_name

missing shortdoc

Syntax:

pi

Predefined constant: \pi in the current precision of the real field.

Syntax:

pi;

real

pi

differentials	hdx, gdx
integer	r	number of iterated applications