Polynomial Maps with Constants on Matrix Algebra
Prachi Saini, Anupam Singh

TL;DR
This paper investigates the surjectivity of specific polynomial maps on matrix algebras, providing complete classifications for matrices of size 3 and 4 based on algebraic properties.
Contribution
It extends previous work by characterizing surjectivity conditions for polynomial maps with constants on matrix algebras of size 3 and 4, relating them to matrix nullity.
Findings
For n=2, surjectivity is fully characterized (from prior work).
For n=3,4, surjectivity depends on the nullity of A_2 and parameters n, k.
The paper provides necessary and sufficient conditions for surjectivity based on algebraic properties.
Abstract
Let be an -algebra and which defines a map by evaluation, called a polynomial map with constant. We consider , the algebra of matrices over an algebraically closed field of characteristic , and polynomial maps given by , where . For , 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 , is invertible, we relate the surjectivity of to the nullity of . When , we completely classify the surjectivity of …
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