Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci

2017/03/16 by Morihiko Saito, Saito, Morihiko · 3 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.1703.05741

Abstract

For a homogeneous polynomial of n variables, we present a new method to compute the roots of Bernstein-Sato polynomial supported at the origin, assuming that general hyperplane sections of the associated projective hypersurface have at most weighted homogeneous isolated singularities. Calculating the dimensions of certain E_r-terms of the pole order spectral sequence for a given integer r∈[2,n], we can detect its degeneration at E_r for certain degrees. In the case of strongly free, locally positively weighted homogeneous divisors on \mathbb P³, we can prove its degeneration almost at E₂ and completely at E₃ together with a symmetry of a modified pole-order spectrum for the E₂-term. These can be used to determine the roots of Bernstein-Sato polynomials supported at the origin, except for rather special cases.

Cited by

• Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points
• Lowest non-zero vanishing cohomology of holomorphic functions
• Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities
• Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables

Related

• Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities
• Bernstein-Sato polynomials of semi-weighted-homogeneous polynomials of nearly Brieskorn-Pham type
• Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points
• Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables
• Bernstein--Sato polynomials of locally quasi-homogeneous divisors in ℂ³