Polynomial-like elements in vector spaces with group actions

TL;DR

This paper investigates polynomial-like elements in vector spaces with group actions, showing that finite-dimensionality of degree-zero elements implies finite-dimensionality for all degrees, linking these elements to periodic solutions of differential equations.

Contribution

It introduces a definition of polynomial-like elements via difference operators and proves a finiteness property for these elements under group actions.

Findings

Polynomial-like elements are characterized as polynomials with periodic coefficients.

Finite-dimensionality of degree-zero polynomial-like elements implies finiteness for all degrees.

Connections are established between polynomial-like elements and solutions of periodic differential equations.

Abstract

In this paper, we study polynomial-like elements in vector spaces equipped with group actions. We first define these elements via iterated difference operators. In the case of a full rank lattice acting on an Euclidean space, these polynomial-like elements are exactly polynomials with periodic coefficients, which are closely related to solutions of periodic differential equations. Our main theorem confirms that if the space of polynomial-like elements of degree zero is of finite dimension then for any $n \in \mathbb{Z}_+$, the space consisting of all polynomial-like elements of degree at most $n$ is also finite dimensional.