Images of Polynomial Maps with Constants on Algebras

Abstract

This thesis studies polynomial maps with constants evaluated on algebras, with particular emphasis on surjectivity and classification of images. Given an algebra $\mathcal A$ and a noncommutative polynomial $\omega$ whose coefficients lie in $\mathcal A$, the associated evaluation map $\omega \colon \mathcal A^{m} \to \mathcal A$ naturally raises several fundamental questions: when is the map surjective, when is its image a vector space, and how do these properties depend on the choice of coefficients? The main focus of this thesis is on diagonal polynomial maps and L'vov-Kaplansky type maps on central simple algebras, such as matrix algebras and quaternion algebras, as well as on the octonion algebra, which is non-associative. In particular, we address the following problems. (1) Surjectivity of diagonal maps. Let $m$ be a positive integer and $n \geq 2$. Given integers $k_1, k_2, \ldots, k_m \geq 1$ and nonzero elements $A_1, \ldots, A_m \in \mathcal A$, consider the diagonal map \begin{align*} \omega \colon M_n(\mathcal A)^m & \longrightarrow M_n(\mathcal A), \\ (x_1, \ldots, x_m) & \mapsto A_1 x_1^{k_1} + \cdots + A_m x_m^{k_m}. \end{align*} We study the minimum value of $m$ for which $\omega$ is surjective. In the case $m = 2$, we determine conditions on $A_1$ and $A_2$ that ensure surjectivity. (2) L'vov-Kaplansky type maps. Let $\mathbb F$ be a field and $\mathcal A = M_n(\mathbb F)$. For \[ \omega = A_1(x_1x_2) - A_2(x_2x_1) \in \mathcal A\langle x_1, x_2\rangle, \] with $A_1, A_2 \in \mathcal A$, we determine precisely when the image of the associated map is a vector space. These problems unify themes arising from Waring-type problems, polynomial identities, and orbit classification under the action of automorphism groups, and contribute to the broader program of understanding the images of polynomial maps on algebras. To address these questions, we employ tools such as canonical forms of matrices, actions of automorphism groups, simultaneous conjugation, and the reduction of solvability over extension fields to solvability over base fields. We first study diagonal maps on matrix algebras over sufficiently large finite fields, algebraically closed fields, and the real field with scalar coefficients, determining the minimum number of variables required for surjectivity. As a consequence, we also describe the images of such maps on Hamiltonian quaternions and division octonion algebras. Using the theory of central simple algebras, we then analyze diagonal maps with coefficients from the algebra itself on $M_2(\overline{\mathbb F})$. By classifying orbit representatives under the action of the automorphism group, we obtain explicit conditions on the coefficients that guarantee surjectivity. Next, we extend this approach to the split octonion algebra, obtained via the Cayley-Dickson construction. Using the classification of orbit representatives under the action of the exceptional group $G_2$, we determine conditions ensuring surjectivity of diagonal maps in two variables over an algebraically closed field. Finally, we classify L\'vov-Kaplansky type maps on $M_2(\overline{\mathbb F})$, determining exactly when their images form vector spaces. Together, these results contribute to a systematic understanding of polynomial images on matrix algebras and related algebraic structures.