Singular Manual: RiemannRochHess

| #### D.5.7.1 RiemannRochHess Procedure from library `hess.lib` (see hess_lib). 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. Example:
    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 (    list Infin = maxorderInfinite(f_aff);) hess.l\
       ib::RiemannRochHess:171
    ==> [1]:
    ==>    [1]:
    ==>       3x5+x3y2-4x2y3-x3y+4x2y2+3y4-6y3+3y2
    ==>    [2]:
    ==>       5x3y2-5x2y3-2x3y+5x2y2
    ==>    [3]:
    ==>       5x4y-5x3y2+3x3y
    ==>    [4]:
    ==>       5x5-5x4y+3x3y
    ==>    [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]:
    ==>       3x5+x3y2-4x2y3-x3y+4x2y2+3y4-6y3+3y2
    ==>    [2]:
    ==>       5x3y2-5x2y3-2x3y+5x2y2
    ==>    [3]:
    ==>       5x4y-5x3y2+3x3y
    ==>    [4]:
    ==>       5x5-5x4y+3x3y
    ==>    [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]:
    ==>       2x5-x3y2-x2y3+x2y2+2y4-4y3+2y2
    ==>    [2]:
    ==>       5x3y2-5x2y3-2x3y+5x2y2
    ==>    [3]:
    ==>       5x4y-5x3y2+3x3y
    ==>    [4]:
    ==>       5x5-5x4y+3x3y
    ==>    [5]:
    ==>       x5+2x4y-3xy4+6xy3-3xy2
    ==> [2]:
    ==>    x5

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