Deep congruences + the Brauer-Nesbitt theorem

TL;DR

The paper establishes a deep connection between mod-$p$ polynomial congruences and $p$-power congruences of roots' power-sum functions, extending classical results and refining the Brauer-Nesbitt theorem for linear operators.

Contribution

It introduces a combinatorial approach using $p$-equivalence on partitions to relate polynomial congruences to root functions, generalizing to torsion-free $bZ_{(p)}$-algebras.

Findings

Proves equivalence of mod-$p$ polynomial and $p$-power root congruences.

Introduces a $p$-equivalence relation on partitions for linear combinations of power-sum functions.

Refines the Brauer-Nesbitt theorem for a single linear operator.

Abstract

We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo divided-power ideals. Our approach is combinatorial: we introduce a $p$-equivalence relation on partitions, and use it to prove that certain linear combinations of power-sum functions are $p$-integral. We also include a second proof, short and algebraic, suggested by an anonymous referee. As a corollary we obtain a refinement of the Brauer-Nesbitt theorem for a single linear operator, motivated by the study of Hecke modules of mod-$p$ modular forms.