A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Translation invariant linear spaces of polynomials

Volume 260 / 2023

Gergely Kiss, Miklós Laczkovich Fundamenta Mathematicae 260 (2023), 163-179 MSC: Primary 32A08; Secondary 13C05. DOI: 10.4064/fm140-10-2022 Published online: 21 November 2022

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.

Authors

  • Gergely KissRényi Institute
    Budapest, Hungary
    e-mail
  • Miklós LaczkovichELTE Eötvös Loránd University
    Budapest, Hungary
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image