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CorrespondenceScrolls :: multiHilbertPolynomial

multiHilbertPolynomial -- Multi-graded Hilbert polynomial for a product of projective spaces

Synopsis

Description

Let M be a module over a polynomial ring P = kk[x0,0..x0,a0..xn-1,0..xn-1,an-1] graded with degree xi,j = ei, the i-th unit vector. If M = Pm is free, then the Hilbert polynomial is the product of the shifted binomiral coefficients binomial(ai+mi+t,ai). In general, the routine computes a free resolution of the coker of the initial matrix of a presentation matrix, and then makes an alternating sum of the Hilbert polynomials of the free modules in the resolution. The polynomial returned has variables hi (the default) or namei if VariableName => "name" is given.

i1 : P = productOfProjectiveSpaces{1,2}

o1 = P

o1 : PolynomialRing
i2 : M1 = P^1

      1
o2 = P

o2 : P-module, free
i3 : multiHilbertPolynomial M1

     1   2   3       1 2        3
o3 = -s s  + -s s  + -s  + s  + -s  + 1
     2 0 1   2 0 1   2 1    0   2 1

o3 : QQ[s , s ]
         0   1

Caveat

Because of the computation of a free resolution, this might be slow on large examples.

See also

Ways to use multiHilbertPolynomial :