Resultant, discriminant
type: Item
instance of: scholarly article
title: Resultant, discriminant
zbMATH Open document ID: 1283.13007
DOI: 10.4171/LEM/58-3-5
author: Michel Demazure
published in: L'Enseignement Mathématique. 2e Série
publication date: 2013-04-03T00:00:00Z
review text: The paper under review is a revised version of a text on the elementary theory of resultants and discriminants intended as an appendix to Chapter IV (``Polynomials and Rational Fractions'') of Bourbaki's treatise on ``Algebra''. The Bourbaki ``congress'' held in july 1969 decided to abandon the project, mainly because some of the results used in the text, like Gauss' lemma for example, depended on the notion of factorial ring which appears only in Chapter VII of Bourbaki's book on ``Commutative Algebra''. Compared to the original version, the paper under review contains some modifications and the addition of two appendices: one on Gauss' lemma and the other one on Hilbert's Nullstellensatz. After this information, extracted from the Introduction of the paper, let us briefly describe its content. In Section 1 one proves the basic result of elimination theory. Let \(A\) be an (always commutative) ring and \(I\) a homogeneous ideal of the polynomial ring \(A[X_1,\dots ,X_n]\). The \textit{eliminating ideal} of \(I\) is the ideal \(\mathfrak{e}(I)\) of \(A\) defined by: \[ \mathfrak{e}(I):= \{a\in A\, | \, aX_i^m \in I\;\text{for}\;1\leq i \leq n \;\text{and}\;m >> 0\}\, . \] If \(k\) is a field and \(\rho: A \rightarrow k\) is a morphism of rings then the \textit{Elimination Theorem} asserts that \(\rho(\mathfrak{e}(I)) = (0)\) if and only if there exists a field extension \(K\) of \(k\) such that the homogeneous elements of \(I\) have a non-trivial common zero in \(K^n\). Actually, if \(\rho(\mathfrak{e}(I)) = (0)\) then one can find such a \(K\) of finite degree over \(k\). In Section 2 one introduces some notation. For fixed integers \(n\geq 1\), \(r\geq 1\), \(d_1\geq 1,\dots ,d_r\geq 1\), one considers the \textit{universal} homogeneous polynomials in \(n\) indeterminates, of degrees \(d_1,\dots ,d_r\), respectively: \[ P_j = { \sum_{| \alpha | = d_j}}T_{\alpha,j}X^\alpha ,\;j = 1,\dots ,r\, , \] where, for \(\alpha = (a_1,\dots ,a_n)\in {\mathbb N}^n\), one denot…
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13P15
Mathematics Subject Classification ID: 13A05
zbMATH DE Number: 6149827
zbMATH Keywords: Polynomials over commutative rings
zbMATH Keywords: elimination theory
zbMATH Keywords: resultant
zbMATH Keywords: discriminant
reviewed by: Iustin Coandă
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/lem/58-3-5
OpenAlex ID: W4246066362
description: scientific article; zbMATH DE number 6149827