Real Jacobian mates
type: Item
instance of: scholarly article
title: Real Jacobian mates
zbMATH Open document ID: 1357.14072
author: Janusz Gwoździewicz
published in: Annales Polonici Mathematici
publication date: 2016-11-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1505.04967
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14P99
zbMATH DE Number: 6654778
zbMATH Keywords: real Jacobian conjecture
zbMATH Keywords: Newton polygon
zbMATH Keywords: real Jacobian mate
MaRDI profile type: Publication
OpenAlex ID: W1775241816
arXiv ID: 1505.04967
cites work: Q3996250
cites work: The real Jacobian conjecture on \(\mathbb R^2\) is true when one of the components has degree 3
DOI: 10.4064/AP3740-8-2016
review text: A polynomial \(q\in \mathbb{R}[x,y]\) is a real Jacobian mate of \(p\in \mathbb{R}[x,y]\) if the Jacobian determinant \(\text{Jac}(p,q)=\frac{\partial p}{ \partial x}\frac{\partial q}{\partial y}-\frac{\partial p}{\partial y}\frac{\partial q}{\partial x}\) vanishes nowhere. The paper gives a wide class of polynomials \(p,\) characterized in terms of the Newton polygon \(\Delta (p)\) of \(p,\) which do not have real Jacobian mate. Namely, if \(\Delta (p)\) has an edge \(S\) such that:NEWLINENEWLINE1. a normal vector \(\vec{v}=(v_{1},v_{2})\) to \(S\) pointing outwards from \( \Delta (p)\) satisfies \(v_{1}>0\) and \(v_{2}/v_{1}<0\),NEWLINENEWLINE2. one endpoint of \(S\) is \((0,1)\),NEWLINENEWLINE3. the curve \(p=0\) has a real branch at infinity associated to \(S\),NEWLINENEWLINEthen \(p\) have no real Jacobian mate. For item 3 it suffices that only endpoints of \(S\) are lattice points, i.e., \(\mathbb{N}_{0}^{2}\cap S=\{\text{two endpoints}\}\).
reviewed by: Tadeusz Krasiński
description: scientific article; zbMATH DE number 6654778