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
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i4 : extIsOnePolynomial coker random(R2^{0,1},R2^{3:-1})
o4 = (3z - 2, false)
o4 : Sequence
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The object extIsOnePolynomial is a method function.