Characterization of polynomials by their invariance properties

TL;DR

This paper characterizes polynomials in multiple variables through their invariance under specific classical groups, providing a new perspective on polynomial symmetry and invariance properties in real and complex fields.

Contribution

It introduces a novel characterization of polynomials as invariant subspaces under group actions, extending classical invariance concepts to broader algebraic settings.

Findings

Polynomials are characterized by invariance under classical groups.

Extension of polynomial invariance characterization to fields with characteristic zero.

Provides a group-theoretic perspective on polynomial structure.

Abstract

We prove that certain classical groups $G\subseteq {\rm GL}(d,\mathbb{R}^d)$ serve to characterize ordinary polynomials in $d$ real variables as elements of finite-dimensional subspaces of $C(\mathbb{R}^d)$ that are invariant by changes of variables induced by translations and elements of $G$. We also show that, if the field $\mathbb{K}$ has characteristic $0$, the elements of $\mathbb{K}[x_1,\cdots,x_d]$ admit a similar characterization for $G= {\rm GL}(d,\mathbb{K})$.