Translation invariant linear spaces of polynomials

A set of polynomials M is called a submodule of C[x1, . . . , xn] if M is a translation invariant linear subspace of C[x1, . . . , xn]. We present a description of the submodules of C[x, y] in terms of a special type of submodules. We say that the submodule M of C[x, y] is an Lmodule of order s if, whenever F (x, y) = ∑N n=0 fn(x) · y n ∈ M is such that f0 = . . . = fs−1 = 0, then F = 0. We show that the proper submodules of C[x, y] are the sums Md+M , where Md = {F ∈ C[x, y] : deg xF < d}, and M is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers. A submodule M ⊂ C[x1, . . . , xn] is decomposable if it is the sum of finitely many proper submodules of M . Otherwise M is indecomposable. It is easy to see that every submodule of C[x1, . . . , xn] is the sum of finitely many indecomposable submodules. In C[x, y] every indecomposable submodule is either an L-module or equals Md for some d. In the other direction we show that Md is indecomposable for every d, and so is every L-module of order 1. Finally, we prove that there exists a submodule of C[x, y] (in fact, an L-module of order 1) which is not relatively closed in C[x, y]. This answers a problem posed by L. Székelyhidi in 2011.