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Translation invariant linear spaces of polynomials
- Source :
- Fundamenta Mathematicae. 260:163-179
- Publication Year :
- 2023
- Publisher :
- Institute of Mathematics, Polish Academy of Sciences, 2023.
-
Abstract
- A set of polynomials $M$ is called a {\it submodule} of $\mathbb{C} [x_1, \dots, x_n ]$ if $M$ is a translation invariant linear subspace of $\mathbb{C} [x_1, \dots, x_n ]$. We present a description of the submodules of $\mathbb{C} [x,y]$ in terms of a special type of submodules. We say that the submodule $M$ of $\mathbb{C} [x,y]$ is an {\it 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 $f_0 =\ldots = f_{s-1}=0$, 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 \textit{deg}_x F<br />Comment: 22 pages
Details
- ISSN :
- 17306329 and 00162736
- Volume :
- 260
- Database :
- OpenAIRE
- Journal :
- Fundamenta Mathematicae
- Accession number :
- edsair.doi.dedup.....99b4d58752d27b9e0046db4526ad9870