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D.9.1.4 Weierstrass
Procedure from library brnoeth.lib (see brnoeth_lib).
- Usage:
Weierstrass( i, m, CURVE ); i,m integers and CURVE a list
- Return:
list WS of two lists:
WS[1] list of integers (Weierstr. semigroup of the curve at place i up to m) WS[2] list of ideals (the associated rational functions)
- Note:
The procedure must be called from the ring CURVE[1][2], where CURVE is the output of the procedure
NSplaces.
i represents the place CURVE[3][i].
Rational functions are represented by numerator/denominator in form of ideals with two homogeneous generators.- Warning:
The place must be rational, i.e., necessarily CURVE[3][i][1]=1.
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); def R=C[1][2]; setring R; // Place C[3][1] has degree 1 (i.e it is rational); list WS=Weierstrass(1,7,C); → Vector basis successfully computed // the first part of the list is the Weierstrass semigroup up to 7 : WS[1]; → [1]: → 0 → [2]: → 3 → [3]: → 5 → [4]: → 6 → [5]: → 7 // and the second part are the corresponding functions : WS[2]; → [1]: → _[1]=1 → _[2]=1 → [2]: → _[1]=y → _[2]=z → [3]: → _[1]=xy → _[2]=z2 → [4]: → _[1]=y2 → _[2]=z2 → [5]: → _[1]=y3 → _[2]=xz2 printlevel=plevel; |
See also: Adj_div; BrillNoether; NSplaces.