Surjectivity of polynomial maps on Matrices
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913337236783104 |
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| author | Panja, Saikat Saini, Prachi Singh, Anupam |
| author_facet | Panja, Saikat Saini, Prachi Singh, Anupam |
| contents | For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=δ_1 X_1^{k_1} + δ_2 X_2^{k_2} + \cdots +δ_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K= \mathbb F_q$ for large enough $q$. Moreover, when $\mathbb K= \mathbb R$ and $m=2$ it is surjective except when $n$ is odd, $k_1, k_2$ are both even, and $δ_1δ_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of Hamiltonian quaternions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_19731 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Surjectivity of polynomial maps on Matrices Panja, Saikat Saini, Prachi Singh, Anupam Group Theory Rings and Algebras 16S50, 11P05 For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=δ_1 X_1^{k_1} + δ_2 X_2^{k_2} + \cdots +δ_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K= \mathbb F_q$ for large enough $q$. Moreover, when $\mathbb K= \mathbb R$ and $m=2$ it is surjective except when $n$ is odd, $k_1, k_2$ are both even, and $δ_1δ_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of Hamiltonian quaternions. |
| title | Surjectivity of polynomial maps on Matrices |
| topic | Group Theory Rings and Algebras 16S50, 11P05 |
| url | https://arxiv.org/abs/2305.19731 |