Polynomial-like elements in vector spaces with group actions

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.