Bernstein-Sato theory modulo $p^m$

Eamon Quinlan-Gallego; Thomas Bitoun

arxiv: 2401.07082 · v3 · pith:CEZGZQ32new · submitted 2024-01-13 · 🧮 math.AC · math.AG· math.NT

Bernstein-Sato theory modulo p^m

Thomas Bitoun , Eamon Quinlan-Gallego This is my paper

Pith reviewed 2026-05-24 04:01 UTC · model grok-4.3

classification 🧮 math.AC math.AGmath.NT

keywords Bernstein-Sato polynomialsZ/p^m coefficientsrational rootsp-torsion strengthmodulo p reductioncharacteristic zero lifting

0 comments

The pith

The Bernstein-Sato polynomial for polynomials over Z/p^m has only rational roots whose negative values match the mod-p reduction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Bernstein-Sato polynomial attached to a polynomial whose coefficients lie in the ring Z/p^m. The definition is required to reduce to the standard theory when m equals one. It then proves that every root of this polynomial is a rational number. The negative roots coincide with the roots of the polynomial obtained after reducing coefficients modulo p. The construction also equips roots with a strength measured by the order of p-torsion they carry and shows that roots of high strength arise by reducing a characteristic-zero Bernstein-Sato polynomial modulo p.

Core claim

For any polynomial f with coefficients in Z/p^m the associated Bernstein-Sato polynomial b_f(s) has only rational roots. The negative roots of b_f(s) are identical to the negative roots of the Bernstein-Sato polynomial of the reduction of f modulo p. Roots carrying sufficiently high p-torsion strength are obtained by reducing Bernstein-Sato roots that exist in characteristic zero.

What carries the argument

The Bernstein-Sato polynomial over Z/p^m, constructed so that it satisfies a functional equation reducing compatibly to the mod-p case and carries a p-torsion strength invariant on its roots.

If this is right

Negative roots of the Z/p^m Bernstein-Sato polynomial agree exactly with those of its mod-p reduction.
Roots may be positive, in contrast to the characteristic-zero theory.
Roots of high p-torsion strength arise by reduction from characteristic-zero roots.
The construction preserves rationality of all roots while extending the theory to positive characteristic with torsion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The strength filtration on roots may allow some characteristic-zero roots to be recovered from finite-characteristic data by lifting through successive p-powers.
The same construction could be used to compare singularity invariants of hypersurfaces across mixed-characteristic settings.

Load-bearing premise

A Bernstein-Sato polynomial can be defined for polynomials with Z/p^m coefficients in a manner compatible with the existing theory when m equals 1.

What would settle it

An explicit polynomial f over Z/p^2 together with a direct computation of its Bernstein-Sato polynomial showing either an irrational root or a negative root different from the one obtained after reduction modulo p.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines Bernstein-Sato polynomials over Z/p^m Z that reduce correctly mod p, proves the roots are rational with negative ones matching the mod p case, shows positive roots occur, and introduces p-torsion strength to recover char 0 roots.

read the letter

The main advance is a definition of the Bernstein-Sato polynomial for polynomials with Z/p^m coefficients that stays compatible when m=1. The roots are rational, the negative roots agree with the mod p reduction, positive roots appear in examples, and strength measured by p-torsion lets strong roots reduce to roots in characteristic zero. The construction is new and the examples make the positive-root claim concrete rather than abstract. The strength notion is a direct way to link the two settings without extra parameters. The soft spots are limited to the fact that the abstract gives no proof outline, so the exact functional equation and the steps to rationality need checking in the body; nothing in the stated claims looks circular or dependent on fitted data. The positive-root examples and the torsion reduction are the parts that stand out as useful. This is for people already working in D-modules or commutative algebra over mixed-characteristic rings. A reader who knows the classical theory will see the value in the examples and the strength construction. It deserves a serious referee because the results are precise, the claims are internally consistent, and the topic addresses a clear gap for arithmetic applications.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a Bernstein-Sato polynomial for polynomials with coefficients in Z/p^m Z that is compatible with the existing theory when m=1. It proves that the roots are rational, that the negative roots agree with those of the mod-p reduction, provides examples showing that roots may be positive, introduces a notion of 'strength' for roots measured by p-torsion, and shows that strong roots lift to roots in characteristic zero via mod-p reduction.

Significance. If the construction and proofs hold, the work extends Bernstein-Sato theory to a p-adic setting, introduces a novel strength invariant that bridges positive and zero characteristics, and supplies the first examples of positive roots. These features could influence the study of singularities and D-modules over rings of mixed characteristic. The compatibility with the m=1 case and the rationality result are particularly useful.

minor comments (2)

The abstract states the main theorems but the manuscript would benefit from a brief outline of the key steps in the definition of the Bernstein-Sato polynomial over Z/p^m Z (e.g., how the functional equation is adapted) already in the introduction.
Notation for the new 'strength' invariant should be introduced with a dedicated symbol and a short comparison table to the classical roots when m=1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending Bernstein-Sato theory to mixed characteristic, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a Bernstein-Sato polynomial over Z/p^m coefficients in a manner stated to be compatible with the m=1 case, then proves rationality of roots, agreement of negative roots with the mod-p case, existence of positive roots, and a p-torsion strength notion that lifts to characteristic zero. No equations, reductions, or self-citations are visible that would make any claimed result equivalent to its inputs by construction. The derivation chain consists of independent definitions followed by proofs, with no fitted parameters renamed as predictions or uniqueness theorems imported from prior self-work in a load-bearing way. This is the expected outcome for a self-contained theoretical construction in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central addition is a new definition whose well-definedness and compatibility rest on domain assumptions from existing Bernstein-Sato theory; no free parameters or invented entities beyond the strength notion are visible in the abstract.

axioms (1)

domain assumption A Bernstein-Sato polynomial exists and satisfies standard functional equations when coefficients are reduced modulo p (m=1 case).
Invoked to ensure the new construction is compatible with existing theory.

invented entities (1)

strength of a root no independent evidence
purpose: Quantify p-torsion to select roots that survive reduction to characteristic zero.
New auxiliary notion introduced to state the final claim about characteristic-zero roots.

pith-pipeline@v0.9.0 · 5642 in / 1226 out tokens · 38916 ms · 2026-05-24T04:01:19.922667+00:00 · methodology