Returns the list of projective Hilbert polynomials of linear spaces summing up to the input polynomial:
i1 : QQ[t]; |
i2 : hp = projectiveHilbertPolynomial(3*t+4) o2 = P + 3*P 0 1 o2 : ProjectiveHilbertPolynomial |
i3 : gD = gotzmannDecomposition hp o3 = {P , - P + P , - 2*P + P , P , P , P , P } 1 0 1 0 1 0 0 0 0 o3 : List |
i4 : sum gD o4 = P + 3*P 0 1 o4 : ProjectiveHilbertPolynomial |
The decomposition suggests the most degenerate geometric object with the given Hilbert polynomial.
i5 : R = QQ[x,y,z,w]; |
i6 : completeIntersection22 = ideal(random(2,R),random(2,R)); o6 : Ideal of R |
i7 : hp = hilbertPolynomial completeIntersection22 o7 = - 4*P + 4*P 0 1 o7 : ProjectiveHilbertPolynomial |
i8 : gD = gotzmannDecomposition hp o8 = {P , - P + P , - 2*P + P , - 3*P + P , P , P } 1 0 1 0 1 0 1 0 0 o8 : List |
The degree of hp is 1, so it is possible to obtain hp as Hilbert polynomial of a scheme in the plane. Gotzmann's decomposition has 4 terms of degree 1 and 2 term of degree 0. This suggests that the generic union of 4 lines and 2 points in a plane should have Hilbert polynomial hp:
i9 : H = random(1,R); |
i10 : fourLines = for i from 1 to 4 list ideal(H,random(1,R)); |
i11 : twoPoints = for i from 1 to 2 list ideal(H,random(1,R),random(1,R)); |
i12 : unionLinesPoints = intersect(fourLines|twoPoints); o12 : Ideal of R |
i13 : hilbertPolynomial unionLinesPoints == hp o13 = true |
The object gotzmannDecomposition is a method function.