Linearizations of binary and ternary forms
type: Item
instance of: scholarly article
title: Linearizations of binary and ternary forms
zbMATH Open document ID: 0631.13007
DOI: 10.1016/0021-8693(87)90171-2
author: Michel Van den Bergh
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: A homogeneous form \(f(x_ 1,...,x_ n)\) of degree \(d\) with coefficients in a field \(F\) has a finite linearization if for some \(m\) there are \(m\times m\) matrices \(\alpha_ 1,...,\alpha_ n\) with entries in \(F\) such that \((\alpha_ 1x_ 1+...+\alpha_ nx_ n)^ d=f(x_ 1,...,x_ n)I\), where \(I\) is the identity matrix. The order \(m\) of \(I\) is called the ``rank'' of the linearization. Finite linearizations of \(f\) corresponding to matrix representations of the generalized Clifford algebra are denoted by \(C_ f\). After proofs of needed propositions, a corollary, lemmas, the author obtains two main theorems: Theorem 1. If \(f\) is a form in two variables and degree \(>3\) such that \(f\) is not divisible by a square then there are irreducible representations of \(C_ f\) of arbitrary high rank. Theorem 2. If \(f\) is a ternary form of degree three such that the projective curve \(f(X_ 1,X_ 2,X_ 3)=0\) is non-singular then \(f\) has a linearization of rank 3.
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 4025558
zbMATH Keywords: homogeneous form
zbMATH Keywords: finite linearization
zbMATH Keywords: matrix representations of the generalized Clifford algebra
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90171-2
OpenAlex ID: W2032753876
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description: scientific article; zbMATH DE number 4025558