Bernstein-Sato theory modulo $p^m$

For fixed prime integer $p>0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z}\mathrm{/}{p}^m$-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.