Bernstein-Sato polynomials of hyperplane arrangements

Abstract

We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than -2 and the multiplicity of the root -1 is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein-Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein-Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.

31 pages

References in corpus (1)

• On real log canonical thresholds

Cited by in corpus (10)

• A young person's guide to mixed Hodge modules
• Hodge spectrum of hyperplane arrangements
• Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials
• Number of Jordan blocks of the maximal size for local monodromies
• Survey on the $D$-module $f^s$
• On b-function, spectrum and multiplier ideals
• Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements
• The Monodromy Conjecture for hyperplane arrangements
• The Bernstein-Sato $b$-function of the Vandermonde determinant
• On meromorphic functions defined by a differential system of order 1, II