Bernstein-Sato theory modulo $p^m$

Abstract: For fixed prime integer $p &gt; 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z} / p<sup>m$-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 ofstrength" for roots by measuring $p$-torsion, and we show that ``strong" roots give rise to roots in characteristic zero through mod-$p$ reduction.