除环上带常数项的多项式映射与广义Kaplansky-L'vov猜想
Polynomial Maps with Constants over Division Algebras and the Generalized Kaplansky--L'vov Conjecture
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中文总结 AI 辅助
该研究针对无限除环上带常数矩阵系数的多项式映射，证明了广义Kaplansky-L'vov猜想对实数域和四元数除环上的2×2矩阵成立，并探讨了这类多项式映射的满射性。
中文摘要 AI 辅助
Kaplansky-L'vov猜想指出，域上全矩阵代数上的多重线性多项式映射的像始终是一个向量空间。尽管该猜想总体上仍未解决，但对于不同域上的2×2和3×3矩阵代数，已取得了实质性进展。最近，Panja、Saini和Singh针对代数闭域上带矩阵系数的多项式映射，提出了广义Kaplansky-L'vov猜想，并验证了其对2×2矩阵的成立性。在本研究中，我们探讨无限除环上带常数矩阵系数的多项式映射的类似问题。具体而言，我们考虑自由代数$M_2(\boldsymbol{D})\boldsymbol{\text{⟨}}x_1,\boldsymbol{\text{…}},x_m\boldsymbol{\text{⟩}}$中的多项式，其形式为$\boldsymbol{\text{ω}}=A_1x_1^{k_1}+\boldsymbol{\text{…}}+A_mx_m^{k_m}$，其中$A_1,\boldsymbol{\text{…}},A_m\boldsymbol{\text{∈}}M_2(\boldsymbol{D})$是固定矩阵，$\boldsymbol{D}$是无限除环，$k_1,\boldsymbol{\text{…}},k_m$是正整数。我们证明了对应的广义Kaplansky-L'vov猜想对$\boldsymbol{R}$上的2×2矩阵及四元数除环$\boldsymbol{H}$成立，同时还研究了这些多项式映射的满射性，这可视为矩阵代数上的广义Waring问题。
英文摘要
The Kaplansky--L'vov conjecture asserts that the image of a multilinear polynomial map on a full matrix algebra over a field is always a vector space. Although the conjecture remains open in general, substantial progress has been made for $2\times 2$ and $3\times 3$ matrix algebras over various fields. Recently, Panja, Saini, and Singh formulated a generalized Kaplansky--L'vov conjecture for polynomial maps with matrix coefficients over algebraically closed fields and verified it for $2\times 2$ matrices. In this work, we investigate an analogous problem for polynomial maps with constant matrix coefficients over an infinite division algebra. Specifically, we consider polynomials in the free algebra $M_2(\mathbb D)\langle x_1,\ldots,x_m\rangle$ of the form $ω= A_1x_1^{k_1}+\cdots+A_mx_m^{k_m},$ where the $A_1,\ldots, A_m\in M_2(\mathbb D)$ are fixed matrices, $\mathbb D$ is an infinite division algebra, and $k_1,\ldots, k_m$ are positive integers. We prove that the corresponding generalized Kaplansky--L'vov conjecture holds for $2\times 2$ matrices over $\mathbb R$ and the quaternion division algebra $\mathbb H$. We also investigate the surjectivity of these polynomial maps. This can be viewed as a generalized Waring problem for matrix algebras.
