Ideals and differential operators in the ring of polynomials of infinitely many variables
type: Item
instance of: scholarly article
title: Ideals and differential operators in the ring of polynomials of infinitely many variables
zbMATH Open document ID: 1349.13043
published in: Periodica Mathematica Hungarica
publication date: 2015-01-07T00:00:00Z
review text: Given an ideal \(J\) of the polynomial ring \(\mathbb{C}[x_1,x_2,\cdots,x_n]\), a theorem due to Krull states that for evey \(f\in \mathbb{C}[x_1,x_2,\cdots,x_n]\setminus J\), there exists a differential operator \(D\) and a complex vector \(c=(c_1,c_2,\cdots,c_n)\) such that \(Dp(c)=0\) for every \(p\in J\) and \(Df(c)\neq 0\). M. Lefranc took profit from this result to prove that the spectral synthesis holds on \(\mathbb{Z}^n\). It is worth noting that the above Krull's theorem does not hold in the polynomial ring \(D:=\mathbb{C}[x_1,x_2,\cdots,x_n,\cdots]\) with countably infinite variables. The paper under review main's goal is to prove that Krull's theorem remains valid in \(D\) via using appropriate generalization of the notion of differential operator. Moreover, the author extends Lefranc's theorem to countable abelian groups by using local polynomials.
reviewed by: Samir Bouchiba
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 6384180
zbMATH Keywords: ideals of rings of polynomials
zbMATH Keywords: polynomials of infinitely many variables
zbMATH Keywords: Krull's theorem
zbMATH Keywords: differential operators
author: M. Laczkovich
MaRDI profile type: Publication
full work available at URL: s10998-014-0059-7
OpenAlex ID: W2060804849
cites work: Q4405003
cites work: Local spectral synthesis on abelian groups
cites work: Harmonic analysis on discrete Abelian groups
cites work: Q3254054
cites work: Q3739243
cites work: Q5824413
DOI: 10.1007/S10998-014-0059-7
description: scientific article; zbMATH DE number 6384180