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# correspondencePolynomial -- computes the Hilbert polynomial of a correspondence scroll

## Synopsis

• Usage:
H = correspondencePolynomial(M,L)
H = correspondencePolynomial(M,L,VariableName =>h)
• Inputs:
• M, ,
• L, a list, of ZZ
• VariableName, ,
• Optional inputs:
• VariableName => ..., default value "s"
• Outputs:
• H, , in QQ[h]

## Description

Let M be a module over a polynomial ring P = kk[x_{0,0}..x_{0,a_0}..x_{n-1,0}..x_{n-1,a_{n-1}}] graded with degree x_{i,j} = e_i, the i-th unit vector, and let b = {b_0..b_{n-1}} be a list of integers. The code computes the multigraded Hilbert polynomial mH(h_0,..,h_{n-1}) and returns H(h) = mH(b_0*h_0, .., b_{n-1}*h_{n-1}).

 i1 : P = productOfProjectiveSpaces {1,1} o1 = P o1 : PolynomialRing i2 : Delta = smallDiagonal P o2 = ideal(- x x + x x ) 0,1 1,0 0,0 1,1 o2 : Ideal of P i3 : M = P^1/(Delta^2) o3 = cokernel | x_(0,1)^2x_(1,0)^2-2x_(0,0)x_(0,1)x_(1,0)x_(1,1)+x_(0,0)^2x_(1,1)^2 | 1 o3 : P-module, quotient of P i4 : correspondencePolynomial (M,{1,1}) o4 = 4s o4 : QQ[s] i5 : correspondencePolynomial (M,{2,2}) o5 = 8s o5 : QQ[s]