Polynomial Maps with Constants on Matrix Algebra
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arXiv
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| Format: | Preprint |
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2026
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| author | Saini, Prachi Singh, Anupam |
| author_facet | Saini, Prachi Singh, Anupam |
| contents | Let $\mathcal A$ be an $\mathbb F$-algebra and $ω\in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A} = M_n(\mathbb{F})$, the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and polynomial maps given by $ω(x_1, x_2) = A_1x_1^k + A_2x_2^k$, where $A_1,A_2\in M_n(\mathbb F)$. For $n=2$, the images of such a map is competely determined in an earlier work (Panja, S.; Saini, P.; Singh, A., Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031). In this article, by assuming one of the coefficients, say $A_1$, is invertible, we relate the surjectivity of $ω$ to the nullity of $A_2$. When $n=3, 4$, we completely classify the surjectivity of $ω(x_1, x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27592 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial Maps with Constants on Matrix Algebra Saini, Prachi Singh, Anupam Rings and Algebras Group Theory Number Theory Primary: 16S50, 11P05, Secondary: 16K20 Let $\mathcal A$ be an $\mathbb F$-algebra and $ω\in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A} = M_n(\mathbb{F})$, the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and polynomial maps given by $ω(x_1, x_2) = A_1x_1^k + A_2x_2^k$, where $A_1,A_2\in M_n(\mathbb F)$. For $n=2$, the images of such a map is competely determined in an earlier work (Panja, S.; Saini, P.; Singh, A., Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031). In this article, by assuming one of the coefficients, say $A_1$, is invertible, we relate the surjectivity of $ω$ to the nullity of $A_2$. When $n=3, 4$, we completely classify the surjectivity of $ω(x_1, x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$. |
| title | Polynomial Maps with Constants on Matrix Algebra |
| topic | Rings and Algebras Group Theory Number Theory Primary: 16S50, 11P05, Secondary: 16K20 |
| url | https://arxiv.org/abs/2604.27592 |