A resultant criterion and formula for the inversion of a polynomial map in two variables
Let k be a field of characteristic 0 and consider a map F: \(k^ n\to k^ n\) defined by polynomials \(F_ 1,...,F_ n\in k[X_ 1,...,X_ n]\). The Jacobian conjecture states that F is invertible (and the inverse is a polynomial map) if and only if \(\det (\partial F_ i/\partial x_ j)\in k^*\). In the paper under review the case \(n=2\) is studied. A criterion in terms of resultants for the invertibility of the map is given. Especially there is an equivalent condition to the Jacobian Umkehrproblem: \(\det (\partial F_ i/\partial X_ j)\in k^*\) implies \((d/dX_ i)R_{X_{3-i}}(F_ 1,F_ 2)\in k^*\), \(i=1,2\). Here \(R_{X_ i}\) is the resultant with respect to \(X_ i\).
scientific article; zbMATH DE number 4154588