Translation invariant linear spaces of polynomials

Kiss, Gergely and Laczkovich, Miklós (2023) Translation invariant linear spaces of polynomials. FUNDAMENTA MATHEMATICAE, 260. pp. 163-179. ISSN 0016-2736

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Abstract

A set of polynomials M is called a {\it 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 {\it L-module of order} s if, whenever F(x,y)=∑Nn=0fn(x)⋅yn∈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]:degxF<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 {\it decomposable} if it is the sum of finitely many proper submodules of M. Otherwise M is {\it 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.

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