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

• 13A35 - Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
• 13N10 - Commutative rings of differential operators and their modules
• 14B05 - Singularities in algebraic geometry
• 14F10 - Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials