Maps preserving zeros of a polynomial

TL;DR

This paper characterizes linear maps on matrix algebras that preserve the zero set of multilinear polynomials, extending understanding of algebraic structure preservation under such maps.

Contribution

It provides a solution for describing zero-preserving linear maps on matrix algebras for general multilinear polynomials under certain conditions.

Findings

Characterized zero-preserving maps for general polynomials on matrix algebras.

Extended results to specific polynomials on broader classes of algebras.

Identified technical conditions under which the preservation property holds.

Abstract

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $\A=M_n(F)$. We also consider quite general algebras $\A$, but only for specific polynomials $f$.