D.9.1.3 BrillNoether

Procedure from library brnoeth.lib (see brnoeth_lib).

Usage:

BrillNoether(G,CURVE); G an intvec, CURVE a list

Return:

list of ideals (each of them with two homogeneous generators, which represent the numerator, resp. denominator, of a rational function).
The corresponding rational functions form a vector basis of the linear system L(G), G a rational divisor over a non-singular curve.

Note:

The procedure must be called from the ring CURVE[1][2], where CURVE is the output of the procedure NSplaces.
The intvec G represents a rational divisor supported on the closed places of CURVE[3] (e.g. G=2,0,-1; means 2 times the closed place 1 minus 1 times the closed place 3).

Example:

LIB "brnoeth.lib";
int plevel=printlevel;
printlevel=-1;
ring s=2,(x,y),lp;
list C=Adj_div(x3y+y3+x);
→ The genus of the curve is 3
C=NSplaces(1..4,C);
// the first 3 Places in C[3] are of degree 1.
// we define the rational divisor G = 4*C[3][1]+4*C[3][3] (of degree 8):
intvec G=4,0,4;
def R=C[1][2];
setring R;
list LG=BrillNoether(G,C);
→ Vector basis successfully computed 
// here is the vector basis of L(G):
LG;
→ [1]:
→    _[1]=1
→    _[2]=1
→ [2]:
→    _[1]=y
→    _[2]=x
→ [3]:
→    _[1]=z
→    _[2]=x
→ [4]:
→    _[1]=y2
→    _[2]=x2
→ [5]:
→    _[1]=xz2+y3
→    _[2]=x3
→ [6]:
→    _[1]=xyz2+y4
→    _[2]=x4
printlevel=plevel;

See also: Adj_div; NSplaces; Weierstrass.