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

Abstract

Let $\hat{\mathcal{O}}$ be a complete local principal ideal ring with residue field $k$ of characteristic not $2$ and $f\in \hat{\mathcal{O}}[x_1,x_2,\dots ,x_m]$. Take $A\in {\mathrm{M}}_n(\hat{\mathcal{O}})$ with its reduction $\overline{A}\in {\mathrm{M}}_n(k)$. In this article, we study the following lifting problem. Suppose there exists a tuple $({\tilde{B}}_1,{\tilde{B}}_2,\dots ,{\tilde{B}}_m)\in {\mathrm{M}}_n(k{)}^m$ of pairwise commuting matrices such that $f({\tilde{B}}_1,{\tilde{B}}_2,\dots ,{\tilde{B}}_m)=\overline{A}$; under what conditions can this solution be lifted to a tuple $(B_1,B_2,\dots ,B_m)\in {\mathrm{M}}_n(\hat{\mathcal{O}}{)}^m$ 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.

v1, 10 pages; comments are welcome