D.15.11.1 RiemannRochHess

Note:

All fractional ideals must represented by a list of size two. The first element is an ideal of k[x,y] and the second element the common denominator, i.e, a polynomial of k[x].

Assume:

The base ring R must be a ring in two variables, say x,y, or three variables, say x,y,z.
If nvars(R) = 2:
- f is an absolutely irreducible polynomial, monic as a polynomial in y.
- List divisorD describes a divisor D of F = Quot(k[x,y]/f). If (s = "ideals")
D is given in ideal representation, i.e., divisorD is a list of size 2.
divisorD[1] is the finite ideal of D, i.e., the
fractional ideal of D of IntCl(k[x],F).
divisorD[2] is the infinite ideal of D, i.e,
the fractional ideal of D of IntCl(k[1/x],F).
If (s = "free")
D is given in free representation, i.e., divisorD is a list of size 2, containing the finite and infinite places of D with exponents.
divisorD[i], i = 1,2, is a list. Each element of the list is again a list. The first entry is a fractional ideal, and the second an integer, the exponent of the place. If nvars(R) = 3:
- f is an absolutely irreducible homogeneous polynomial describing the projective plane curve corresponding to the function field F. We assume that the dehomogenization of f w.r.t. z is monic as a polynomial in y.
List divisorD describes a divisor D of F.
If (s = "ideals")
D is given in ideal representation, i.e., divisorD is a list of size 2. divisorD[1] is an ideal of the base ring representing the positive divisor of D and
divisorD[2] is an ideal of the base ring representing the negative divisor. (i.e. D = (I) -(J)).
If (s = "free")
D is given in free representation, i.e., divisorD is a list of places of D. D[i][1] is an prime ideal and D[i][2] an integer, the exponent of the place.

Return:

A vector space basis of the Riemann-Roch space of D,
stored in a list RRBasis. The list RRBasis contains a list, say rbasis, and a polynomial, say den. The basis of L(D) consists of all rational functions g/den, where g is an element of rbasis.

LIB "hess.lib";
ring R = 0,(x,y),dp;
poly f  = y^2*(y-1)^3-x^5;
list A1 = list(ideal(x,y-1),1),2;
list A2 = list(ideal(y^2-y+1,x),1),3;
list A3 =  list(ideal(1,y-x),x),-2;
list D = A1,A2;
list E = list(A3);
RiemannRochHess(f,list(D,E),"free");
==> // ** redefining Infin **
==> [1]:
==>    [1]:
==>       5x2y3-5xy4-5x2y2+10xy3-2y4-5xy2+4y3-2y2
==>    [2]:
==>       2x3y2+x2y3-3xy4-2x3y-x2y2+6xy3-3xy2
==>    [3]:
==>       5x3y2-5xy4-4x3y+10xy3-5xy2
==>    [4]:
==>       5x4y-5xy4-x3y+10xy3-5xy2
==>    [5]:
==>       x5+2x4y-3xy4+6xy3-3xy2
==> [2]:
==>    x5
ring S = 0,(x,y,z),dp;
poly f = y^2*(y-1)^3-x^5;
f  = homog(f,z);
ideal P1 = x,y-z;
ideal P2 = y^2-yz+z^2,x;
ideal P3 = x-y,z;
list B1 = P1,2;
list B2 = P2,3;
list B3 = P3,-2;
list Ddivisor = B1,B2,B3;
RiemannRochHess(f,Ddivisor,"free");
==> [1]:
==>    [1]:
==>       5x2y3-5xy4-5x2y2+10xy3-2y4-5xy2+4y3-2y2
==>    [2]:
==>       2x3y2+x2y3-3xy4-2x3y-x2y2+6xy3-3xy2
==>    [3]:
==>       5x3y2-5xy4-4x3y+10xy3-5xy2
==>    [4]:
==>       5x4y-5xy4-x3y+10xy3-5xy2
==>    [5]:
==>       x5+2x4y-3xy4+6xy3-3xy2
==> [2]:
==>    x5
ideal I = intersect(P1^2,P2^3);
ideal J = P3^2;
RiemannRochHess(f,list(I,J),"ideals");
==> [1]:
==>    [1]:
==>       5x2y3-5xy4-5x2y2+10xy3-2y4-5xy2+4y3-2y2
==>    [2]:
==>       2x3y2+x2y3-3xy4-2x3y-x2y2+6xy3-3xy2
==>    [3]:
==>       5x3y2-5xy4-4x3y+10xy3-5xy2
==>    [4]:
==>       5x4y-5xy4-x3y+10xy3-5xy2
==>    [5]:
==>       x5+2x4y-3xy4+6xy3-3xy2
==> [2]:
==>    x5