Real Jacobian mates
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., \(\mathbb{N}_{0}^{2}\cap S=\{\text{two endpoints}\}\).
scientific article; zbMATH DE number 6654778