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<h2><a name="AlffDivisorLBasisShort"></a>
AlffDivisorLBasisShort
</h2>

Computes a basis of a Riemann-Roch space.

<p><p>

<h3>
Syntax:
</h3>
<pre>B := AlffDivisorLBasisShort(D);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  B  </pre></td>
<td>of pairs of basis elements b_i and degree bounds d_i</td>
</tr>
<tr>
<td>alff divisor</td>
<td><pre>  D  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AlffDivisorLBasis">AlffDivisorLBasis</a>, <a href="kashref.html#AlffDivisorLDim">AlffDivisorLDim</a>, <a href="kashref.html#AlffDivisorLIndex">AlffDivisorLIndex</a>, <a href="kashref.html#AlffDivisorDeg">AlffDivisorDeg</a>, <a href="kashref.html#AlffGenus">AlffGenus</a>, <a href="kashref.html#Alff">Alff</a>

<p>

<h3>
Description:
</h3>
Let F = k(T,y) be an algebraic function field defined by
f(T,y)=0 over k. Given a divisor D of F this
function returns a basis of the k-vector space  {cal
L}(D) = { a \in F^\times \;|\; (a)  &gt;=  -D } cup { 0 }
 in the short form  B = [ [b_1, d_1], \dots, [b_k,
d_k] ]  with b_i \in F^\times and d_i \in Z^{ &gt;= 
0} for all 1  &lt;=  i  &lt;=  k and with k  &lt;=  n, where n
denotes the degree in y of the defining equation f of
F, such that  {cal L}(D) = \left{ \sum_{i=1}^k
\lambda_i b_i \;|\; \lambda_i \in k[T]  with deg 
\lambda_i  &lt;=  d_i  for  1  &lt;=  i  &lt;=  k right}. 
The basis elements b_i are represented in the finite maximal
order. \par With option <tt> "raw"</tt> a complete list
of n := \deg_y f tuples [b_i, d_i] with d_i \in Z and the
property as above is returned. The algorithm is described in
He2.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> AlffInit(FF(5,1));;
kash> AlffOrders(y^2+T^3+1);;
kash> infty := AlffPlaceSplit(F, 1/T)[1];
Alff place < [ 1/T, 0 ], [ 0, 1 ] >
kash> l := AlffPlaceSplit(F, T+3);
[ Alff place < [ T + 3, 0 ], [ 1, 1 ] >, 
  Alff place < [ T + 3, 0 ], [ 4, 1 ] > ]
kash> D := 2*infty + 3*l[1] - l[2];
Alff divisor
[ [ Alff place < [ 1/T, 0 ], [ 0, 1 ] >, 2 ],
[ Alff place < [ T + 3, 0 ], [ 1, 1 ] >, 3 ],
[ Alff place < [ T + 3, 0 ], [ 4, 1 ] >, -1 ] ]

kash> AlffDivisorLBasisShort(D);
> [ [ [ 4*T^3 + 4*T^2 + 2*T + 1, T ] / (T^3 + 4*T^2 + 2*T + 2), 1 ], 
  [ [ 4*T^3 + 2*T^2 + 4, 1 ] / (T^3 + 4*T^2 + 2*T + 2), 1 ] ]

</pre>

<p>


<hr>

<p>
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