Polynomial Maps with Constants over Division Algebras and the Generalized Kaplansky--L'vov Conjecture
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 $\omega = A_1x_1^{k_1}+\cdots+A_mx_m^{k_m},$ 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.