Subresultants and locally nilpotent derivations.
type: Item
instance of: scholarly article
title: Subresultants and locally nilpotent derivations.
zbMATH Open document ID: 1087.13015
author: M'hammed El Kahoui
published in: Linear Algebra and its Applications
publication date: 2004-03-29T00:00:00Z
review text: For \(P\), \(Q\) univariate polynomials of degree \(p\), resp. \(q\), over a commutative ring with unity, let \(\delta (p,q)\) denote \(\min (p,q)\) if \(p\neq q\) and \(p-1\) otherwise. For \(i\) at most \(\delta (p,q)\), let \(Sr_i (P,Q)\) denote the \(i\)th subresultant polynomial associated to \(P\) and \(Q\). It is proved that \(Sr_i (P,Q)\) is the determinant of a \((p+q-2i)\times (p+q-2i)\) matrix whose entries are taken from the list of coefficients of \(P\), \(P'\), \(\frac{P''}{2}, \ldots, \frac{P^{(p)}}{p!}\), \(Q\), \(Q'\), \(\frac{Q''}{2}, \ldots, \frac{Q^{(q)}}{q!}\). Moreover, the determinants of suitable \((p+q-2i-1)\times (p+q-2i-1)\) matrices provide polynomials \(U_i\) and \(V_i\) such that \(U_iP+ V_iQ=Sr_i (P,Q)\). These properties are established by specialization of similar results stated for Lie series associated to locally nilpotent derivations on commutative rings containing the rationals.
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 13P05
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 13N15
zbMATH DE Number: 2062871
zbMATH Keywords: subresultant sequence
zbMATH Keywords: locally nilpotent derivation
zbMATH Keywords: Bézout identity
MaRDI profile type: Publication
full work available at URL: j.laa.2003.11.004
OpenAlex ID: W2101796132
cites work: Algorithms in real algebraic geometry
cites work: On Euclid's Algorithm and the Theory of Subresultants
cites work: Subresultants and Reduced Polynomial Remainder Sequences
cites work: An elementary approach to subresultants theory.
cites work: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: Spécialisation de la suite de Sturm et sous-résultants (I)
cites work: Q4238962
cites work: Computing Hopf Bifurcations I
cites work: Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens
cites work: Subresultants under composition
cites work: New structure theorem for subresultants
DOI: 10.1016/J.LAA.2003.11.004
Recommended article: A note on locally nilpotent derivations
Recommended article: Local derivations of subrings of matrix rings.
Recommended article: Locally Nilpotent Sets of Derivations
Recommended article: On some properties of locally nilpotent derivations
Recommended article: Derivations and bounded nilpotence index
Recommended article: DERIVATIONS OF THE LOCALLY NILPOTENT MATRIX RINGS
Recommended article: Q5476107
Recommended article: Locally nilpotent derivations on modules
Recommended article: Locally nilpotent derivations and the structure of rings
Recommended article: Locally nilpotent derivations and their applications. III
description: scientific article; zbMATH DE number 2062871