2016/09/15 by Morihiko Saito, Saito, Morihiko · 4 citations
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1609.04801
openalex publication_date 2016/09/15 · openalex created_date 2017/03/23 · openalex updated_date 2026/07/28
We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial f of a projective hypersurface Z⊂\mathbb Pn-1 of degree d having only weighted homogeneous isolated singularities. We prove the E2-degeneration of the pole order spectral sequence so that the computation of roots is reduced to the one of the Hilbert series of the Jacobian ring of f except the special case where f is annihilated by a nonzero vector field on \mathbb Cn with linear function coefficients. In the three variable case with d>4 we may assume that this vector field is a linear combination of x∂x, y∂y, z∂z, where f is called extremely degenerated; in particular, the latter case does not contain any essential indecomposable central hyperplane arrangement in \mathbb C3. Combined with the self-duality of the Koszul complex and a theorem of Dimca and Popescu, it implies for n=3 with d>4 except the extremely degenerated case that Rf=(1)/(d)(\mathbb Z∩[3,k'])∪ RZ. Here Rf,RZ are the roots of Bernstein-Sato polynomials of f and Z up to sign, and k'=max(2d-3,kmax+3) with kmax the maximal degree of the ``torsion part" of the Jacobian ring, where the latter is known to be at most 2d-5 in the hyperplane arrangement case.