Gordan ideals in the theory of binary forms
type: Item
instance of: scholarly article
title: Gordan ideals in the theory of binary forms
zbMATH Open document ID: 0811.13011
author: Jerzy Weyman
published in: Journal of Algebra
publication date: 1995-05-01T00:00:00Z
review text: The author considers subvarieties \(X_{p,q}\) associated with a two dimensional complex vector space \(V\) and binary forms of degree \(g\) having a root of multiplicity \(\geq p\). He started the study of these objects in J. Algebra, 122, No. 1, 244-249 (1989; Zbl 0689.14001). In this paper he finds an explicit deformation of the rings of covariants of subvarieties \(X_{p,g}\) to a ring generated by monomials. There is deduced an explicit estimate for the degrees of generating covariants. It is proved that if \(2k - 2 < g/2\) then the defining ideal \(J_{g - k + 1}\) of \(X_{g - k + 1,g}\) is generated by quadrics. This allows a geometric interpretation of Gordan's algorithm for computing covariants of binary forms [cf. \textit{J. H. Grace} and \textit{A. Young}, ``The algebra of invariants'' (1903)]. Finally there are obtained applications of multiplicity ideals to find the invariants of binary forms of low degrees. The paper contains many suggestive examples.
reviewed by: Doru Ştefănescu
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 554248
zbMATH Keywords: Gordan ideal
zbMATH Keywords: deformation of the rings of covariants of subvarieties
zbMATH Keywords: monomials
zbMATH Keywords: invariants of binary forms
MaRDI profile type: Publication
full work available at URL: jabr.1993.1225
OpenAlex ID: W2080630212
DOI: 10.1006/JABR.1993.1225
description: scientific article; zbMATH DE number 554248