# extIsOnePolynomial -- check whether the Hilbert function of Ext(M,k) is one polynomial

## Synopsis

• Usage:
(p,t) = extIsOnePolynomial M
• Inputs:
• M, , module over a complete intersection
• Outputs:
• p, , p(z)=pe(z/2), where pe is the Hilbert poly of Ext^{even}(M,k)
• t, , true if the even and odd polynomials match to form one polynomial

## Description

Computes the Hilbert polynomials pe(z), po(z) of evenExtModule and oddExtModule. It returns pe(z/2), and compares to see whether this is equal to po(z/2-1/2). Avramov, Seceleanu and Zheng have proven that if the ideal of quadratic leading forms of a complete intersection of codimension c generate an ideal of codimension at least c-1, then the betti numbers of any module grow, eventually, as a single polynomial (instead of requiring separate polynomials for even and odd terms.) This script checks the result in the homogeneous case (in which case the condition is necessary and sufficient.)

 i1 : R1=ZZ/101[a,b,c]/ideal(a^2,b^2,c^5) o1 = R1 o1 : QuotientRing i2 : R2=ZZ/101[a,b,c]/ideal(a^3,b^3) o2 = R2 o2 : QuotientRing i3 : extIsOnePolynomial coker random(R1^{0,1},R1^{3:-1}) 1 2 1 o3 = (-z - -z + 3, true) 2 2 o3 : Sequence i4 : extIsOnePolynomial coker random(R2^{0,1},R2^{3:-1}) o4 = (3z - 2, false) o4 : Sequence