  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.
