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])
|
The object traceMatrix is a method function.