Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings
Saikat Panja, Ayon Roy, Anupam Singh
TL;DR
This work addresses lifting solutions of polynomial equations on matrix algebras from a residue field $k$ to a complete local principal ideal ring $\widehat{\mathscr O}$ with residue field $k$ (char not $2$). The authors develop a Hensel-type lifting framework for matrix-valued polynomial maps, leveraging null ideals and centralizers, and prove that for cyclic $\overline{A}$ the null ideals at each level are principal and generated by successive reductions of a monic $F(t)$. They extend existing length-two results to arbitrary length by an inductive construction and then generalize to a multivariate setting: if $F(x_1,\dots,x_m)$ has commuting inputs and invertible partial derivatives for the first $r$ variables, a lift exists from a residue-field solution to a solution over $\widehat{\mathscr O}$ with $F(B)=A$. The results connect noncommutative lifting to Hensel-type arguments and have potential implications for lifting problems in representation theory and polynomial maps on algebras.
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.