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Abstract

A set of polynomials is called a submodule of $\mathbb C [x_1 ,\ldots , x_n ]$ if it is a translation invariant linear subspace of $\mathbb C [x_1 ,\ldots , x_n ]$. We present a description of the submodules of $\mathbb C [x,y]$ in terms of a special type of submodules. We say that submodule $M$ of $\mathbb C [x,y]$ is an L-module of order $s$ if, whenever $F(x,y)=\sum _{n=0}^N f_n (x) \cdot y^n \in M$ is such that $f_0 =\cdots = f_{s-1}=0$, then $F=0$. We show that the proper submodules of $\mathbb C [x,y]$ are the sums $M_d +M$, where $M_d =\{ F\in \mathbb C [x,y] \colon \deg _x F \lt d\}$, and $M$ is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers.

A submodule $M\subset \mathbb C [x_1 ,\ldots , x_n ]$ is decomposable if it is the sum of finitely many proper submodules of $M$. Otherwise $M$ is indecomposable. In $\mathbb C [x,y]$ every indecomposable submodule is either an L-module or equals $M_d$ for some $d$. In the other direction we show that $M_d$ is indecomposable for every $d$, and so is every L-module of order $1$.

Finally, we prove that there exists a submodule of $\mathbb C [x,y]$ (in fact, an L-module of order $1$) which is not relatively closed in $\mathbb C [x,y]$. This answers a problem posed by L. Székelyhidi in 2011.