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D.10.1.3 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 space 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;
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See also:
Adj_div;
NSplaces;
Weierstrass.
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