traceMatrix -- Compute trace matrix.

Synopsis

• Usage:
traceMatrix(I,ib)
• Inputs:
• I, an ideal, the homogeneous ideal I of an irreducible plane algebraic curve C over K
• ib, , containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K given as a row matrix of length n=degree(I).
• Outputs:
• , the trace matrix of the integral basis ib.

Description

Computes the trace matrix for the integral basis ib, i.e., the matrix containing the traces of the elements ib_(0,i)*ib_(0,j).

We assume that I has the following properties:

Denote the variables of R=ring(I) by v,u,z. All singularities of C have to lie in the chart z!=0 and the curve should not contain (1:0:0).

We assume that ib has the following properties: The entries are in K(u)[v] inside frac(R) where the i-th entry is of degree $i$ in v for i=0..n-1. Note that this always can be achieved.

A rational curve with three double points:

 i1 : K=QQ o1 = QQ o1 : Ring i2 : R=K[v,u,z] o2 = R o2 : PolynomialRing i3 : I=ideal(v^4-2*u^3*z+3*u^2*z^2-2*v^2*z^2) 4 3 2 2 2 2 o3 = ideal(v - 2u z - 2v z + 3u z ) o3 : Ideal of R i4 : Rvu=K[v,u]; i5 : QR=frac(Rvu); i6 : ib=matrix {{1, v, (-1+v^2)/(-1+u), 1/(-1+u)/u*v^3+(-2+u)/(-1+u)/u*v}}; 1 4 o6 : Matrix QR <--- QR i7 : traceMatrix(I,ib) o7 = | 4 0 0 0 | | 0 4 0 8u-4 | | 0 0 8u+4 0 | | 0 8u-4 0 16 | 4 4 o7 : Matrix (frac QQ[u]) <--- (frac QQ[u])

The Cusp:

 i8 : K=QQ; i9 : R=K[v,u,z]; i10 : I=ideal(v^3-u^2*z) 3 2 o10 = ideal(v - u z) o10 : Ideal of R i11 : Rvu=K[v,u]; i12 : QR=frac(Rvu); i13 : ib=matrix({{1,v,v^2/u}}); 1 3 o13 : Matrix QR <--- QR i14 : traceMatrix(I,ib) o14 = | 3 0 0 | | 0 0 3u | | 0 3u 0 | 3 3 o14 : Matrix (frac QQ[u]) <--- (frac QQ[u])

A curve of genus 4:

 i15 : K=QQ; i16 : R=K[v,u,z]; i17 : I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3); o17 : Ideal of R i18 : Rvu=K[v,u]; i19 : QR=frac(Rvu); i20 : ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}}); 1 6 o20 : Matrix QR <--- QR i21 : traceMatrix(I,ib) o21 = | 6 0 0 | 0 0 0 | 0 0 (-28u4-84u3-84u2-28u)/5 | 0 (-28u3-56u2-28u)/5 0 | (-28u2-56u-28)/5 0 (-36u4-36u3)/5 | 0 (-36u3-36u2)/5 0 ----------------------------------------------------------------------- 0 (-28u2-56u-28)/5 (-28u3-56u2-28u)/5 0 0 (-36u4-36u3)/5 -36u4/5 0 0 (196u4+784u3+1176u2+784u+196)/25 (196u4+980u3+1372u2+588u)/25 0 ----------------------------------------------------------------------- 0 | (-36u3-36u2)/5 | 0 | (196u4+980u3+1372u2+588u)/25 | 0 | (420u4+1764u3+1260u2-84u)/25 | 6 6 o21 : Matrix (frac QQ[u]) <--- (frac QQ[u])