Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings

TL;DR

This paper investigates conditions under which solutions to polynomial equations on matrices over residue fields can be lifted to solutions over complete local principal ideal rings, especially focusing on cyclic matrices and Hensel-like hypotheses.

Contribution

It establishes criteria for lifting solutions of polynomial equations on matrices from residue fields to complete local rings, extending classical Hensel lemma concepts to matrix equations.

Findings

Lifting is always possible for cyclic matrices under certain hypotheses.

Provides conditions analogous to Hensel lemma for matrix polynomial equations.

Addresses the problem of commuting matrices satisfying polynomial relations.

Abstract

Let $\widehat{\mathscr O}$ be a complete local principal ideal ring with residue field $k$ of characteristic not $2$ and $f\in \widehat{\mathscr O}[x_1,x_2,\dots,x_m]$. Take $A\in \mathrm M_n(\widehat{\mathscr O})$ with its reduction $\overline{A}\in \mathrm M_n(k)$. In this article, we study the following lifting problem. Suppose there exists a tuple $(\widetilde{B}_1, \widetilde{B}_2, \dots,\widetilde{B}_m)\in \mathrm M_n(k)^m$ of pairwise commuting matrices such that $f(\widetilde{B}_1, \widetilde{B}_2, \dots,\widetilde{B}_m) = \overline{A}$; under what conditions can this solution be lifted to a tuple $(B_1,B_2,\dots,B_m)\in \mathrm M_n(\widehat{\mathscr O})^m$ of pairwise commuting matrices satisfying $f(B_1,B_2,\dots,B_m)=A$? For $\overline{A}$ cyclic, we show that, under suitable hypotheses analogous to those appearing in Hensel lemma, such a lifting is always possible.