Detection of special curves via the double resultant
type: Item
instance of: scholarly article
title: Detection of special curves via the double resultant
zbMATH Open document ID: 1391.13055
author: Antoni Ferragut
author: Johanna D. García-Saldaña
author: Armengol Gasull
published in: Qualitative Theory of Dynamical Systems
publication date: 2018-04-03T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/182490
review text: Given a polynomial of degree \(n\), \(F=a_ny^n+\ldots+a_0\), \(a_n\neq 0\), with coefficients in a commutative ring, the discriminant is equal, up to sign, to the quotient of the resultant of \(F\) and \(dF/dy\) by the leading coefficient \(a_n\). It is well known that a nonconstant polynomial over a field has a multiple root if and only if its discriminant is zero. In literature, it is frequently used the double (or iterated) discriminant of multivariate polynomials in order to deal with different problems of algebraic geometry. In this paper, the authors analyze the use of the double resultant instead. Given a bivariate polynomial whose coefficients depend on a parameter \(b\), \(F_b(x,y)\), the double resultant is applied in order to compute the values of \(b\) for which \(F_b(x,y)\) has singularities. Some examples of different nature where the double resultant is used are introduced: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.
reviewed by: Gema Maria Diaz Toca
Mathematics Subject Classification ID: 13P15
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 68U01
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6855267
zbMATH Keywords: resultant
zbMATH Keywords: double resultant
zbMATH Keywords: discriminant
zbMATH Keywords: tame maps
zbMATH Keywords: applications
MaRDI profile type: Publication
OpenAlex ID: W2259835047
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DOI: 10.1007/S12346-015-0180-X
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description: scientific article; zbMATH DE number 6855267