Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings
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arXiv
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2026
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| author | Panja, Saikat Roy, Ayon Singh, Anupam |
| author_facet | Panja, Saikat Roy, Ayon Singh, Anupam |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04576 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings Panja, Saikat Roy, Ayon Singh, Anupam Group Theory Commutative Algebra Rings and Algebras 15B33, 16S50, 16S85 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. |
| title | Lifting solutions of polynomial equations on matrices over field to complete local principal ideal rings |
| topic | Group Theory Commutative Algebra Rings and Algebras 15B33, 16S50, 16S85 |
| url | https://arxiv.org/abs/2602.04576 |