A criterion for the primeness of ideals generated by polynomials with separated variables
type: Item
instance of: scholarly article
title: A criterion for the primeness of ideals generated by polynomials with separated variables
zbMATH Open document ID: 0962.14002
DOI: 10.1007/BF02780169
author: Konrad Neumann
published in: Israel Journal of Mathematics
publication date: 2000-03-01T00:00:00Z
review text: Let \(K\) be an algebraically closed field of characteristic \(p\geq 0\), \(f_i=\sum^n_{j=1} f_{ij}(X_j)\in K[X_1,...,X_n]\), \(i=1,\dots,m\), polynomials with separated variables and \(V\) the closed subscheme of \(\mathbb{A}^n\) defined by \(f_i=0\), \(i=1,\dots,m\). Let \(\mu_j=\max \{\deg(f_{1j}), \dots,\deg (f_{mj})\}\), \(j=1,\dots,n\) and assume that \(p\) does not divide \(\mu_j\) if \(p>0\) and \(\mu_j\neq 0\) if \(p=0\). Consider the matrix \(A=(a_{ij})\), where \(f_{ij}= a_{ij} X_j^{\mu_j}+\cdots\). In the paper under review, the author shows that \(V\) is a variety (i.e., an irreducible and reduced closed subscheme of \(\mathbb{A}^n)\), if each non-trivial linear combination of the rows of \(A\) has at least three non-zero entries.
reviewed by: Tiberiu Dumitrescu
Mathematics Subject Classification ID: 14A10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 1389700
zbMATH Keywords: closed subscheme
zbMATH Keywords: variety
MaRDI profile type: Publication
cites work: Q3239311
cites work: Q4317713
cites work: The red book of varieties and schemes
cites work: Q5734450
cites work: Q3947744
cites work: Q5330665
full work available at URL: https://doi.org/10.1007/bf02780169
OpenAlex ID: W2050090548
description: scientific article; zbMATH DE number 1389700