Bernstein-Sato theory for arbitrary ideals in positive characteristic

Abstract

Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the F F -jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to F F -jumping numbers.

Bernstein–sato Roots for Monomial Ideals in Positive Characteristic

“…Then there is a monomial ideal J whose radical contains a, a rational number β and a sequence e i ∞ of positive integers such that ν J ap (p e i d ) = βp e i d + α. We remark that, by [16,Theorem 6.9], α is in Z (p) and thus we can always find some…”

“…By [16,Thm. 6.9], α is in Z (p) and thus we may find some d > 0 such that α(p d − 1) ∈ N. By replacing d with a multiple, we may also assume that p d ≡ 1 mod M .…”

“…Our proof relies heavily on results from [6]. In Section 2, we review the notion of Bernstein-Sato root as defined in [16] as well as the needed theorems from [6]. We then prove our result in Section 3.…”

“…In [16] the approaches of Mustaţă and Bitoun were expanded to arbitrary ideals and, in particular, a notion of Bernstein–Sato root of is defined by generalizing a previous definition of Bitoun. These Bernstein–Sato roots are characteristic- analogues of the roots of the Bernstein–Sato polynomial (it is a question in [16] whether one can find an analogue for the multiplicity of a root).…”

“…The Bernstein–Sato roots of are negative, rational (i.e., they lie in ) and encode some information about the -jumping numbers of [16]. Furthermore, the definition of the Bernstein–Sato root in prime characteristic is compatible with that of the Bernstein–Sato polynomial in characteristic zero [16, §6.1].…”