Nonlinear maps preserving the polynomial

TL;DR

This paper characterizes nonlinear maps that preserve a polynomial structure, generalizing known results for matrix invariants like determinants, and introduces a new invariant space related to the polynomial's gradient.

Contribution

It provides a comprehensive characterization of nonlinear maps preserving polynomial functions, extending classical linear preservers to nonlinear contexts and introducing the invariant space alL_P.

Findings

Characterization of maps preserving polynomial structures in characteristic zero.

Introduction of the invariant space alL_P with key properties.

Explicit description of nonlinear maps preserving the determinant of rectangular matrices.

Abstract

Let $\mathbb F$ be a field and $P \in \mathbb F [x_1,\ldots, x_n]$ be a homogeneous polynomial such that $|\mathbb F| > \deg(P)$ and $\phi, \psi\colon \mathbb F^n \to \mathbb F^n$ be two maps such that $P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$ for all $\lambda \in \mathbb F$ and $\mathbf{x}, \mathbf{y} \in \mathbb F^n.$ We provide the characterization of all such $\phi$ and $\psi$ for all polynomials in the case if $\mathrm{char}(\mathbb F) = 0$ and for all polynomials satisfying certain condition in the case if $\mathrm{char}(\mathbb F) > 0$. This characterization generalizes the existing results regarding the linear maps on matrices preserving the determinant, the immanant and other homogeneous polynomial functions of matrix entries. To obtain the main result of this paper, we introduce the vector space $\mathcal L_{P} \subseteq {\mathbb F^n}^*$ spanned by the range of the gradient field of $P \in \mathbb F[x_1,\ldots, x_n]$. Being a linear invariant associated with $P,$ this space has several remarkable properties and may also be used for studying the linear maps preserving $P$. In addition, we demonstrate how the main result could be applied to the particular polynomial matrix invariants. Namely, we provide an explicit description of corresponding pairs of nonlinear maps $\phi, \psi$ for the case where $P$ is equal to the Cullis' determinant of $n\times k$ rectangular matrix (with the assumption that $n \ge k + 2$ and $k \ge 3$).