将域上的多项式方程矩阵解提升到完备局部主理想环
Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings
AI总结：
本文研究了如何将域上的多项式方程矩阵解提升到完备局部主理想环，证明在适当条件下，循环矩阵的解可以被提升。
摘要：
设$\widehat{\mathscr O}$是一个具有余域$k$的完备局部主理想环，其特征不为2，$f\in \widehat{\mathscr O}[x_1,x_2,\dots,x_m]$。取$A\in \mathrm M_n(\widehat{\mathscr O})$及其约简$\overline{A}\in \mathrm M_n(k)$。本文研究以下提升问题。假设有元组$(\widetilde{B}_1, \widetilde{B}_2, \dots,\widetilde{B}_m)\in \mathrm M_n(k)^m$的矩阵元组，使得$f(\widetilde{B}_1, \widetilet{B}_2, \dots,\widetilde{B}_m) = \overline{A}$；在什么条件下可以将此解提升到元组$(B_1,B_2,\dots,B_m)\in \mathrm M_n(\widehat{\mathscr O})^m$的矩阵元组，使得$f(B_1,B_2,\dots,B_m)=A$？对于$\overline{A}$为循环矩阵的情况，我们在类似于Hensel引理出现的适当假设下，证明这种提升总是可能的。
英文摘要：
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.
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