Bernstein-Sato theory modulo $p^m$

Abstract

For fixed prime integer $p>0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z}/p^m$-coefficients, compatible with existing theory in the case $m=1$. We show that the ``roots" of such polynomials are rational and we show that the negative roots agree with those of the mod-$p$ reduction. We give examples to show that, surprisingly, roots may be positive in this context. Moreover, our construction allows us to define a notion of ``strength" for roots by measuring $p$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$p$ reduction.

Comments welcome. v3: final version. v2: fixed typos, small changes in notation, and additional example following suggestions from the referee