Jacobian pairs, \(D\)-resultants, and automorphisms of the plane (Q1903694)

The authors prove the following explicit characterization of polynomial automorphisms of the affine plane \(k^2\), \((k\) is an arbitrary field of characteristic \(p)\): Let \(F = (f,g) : k^2 \mapsto k^2\) be a polynomial mapping. Assume that \(p = 0\) or \(p\) does not divide \([k(x,y) : k(f,g)]\). Let \(J \in k [x,y]\) be the jacobian determinant of \(F\) and \(D \in k [x,y]\) be the \(D\)-resultant of \(f\) and \(g\) with respect to \(y\), i.e., \(D(x,y) = \text{Res}_s ({f(x,s) - f(x,y) \over s - y},\;{g(x,s) - g(x,y) \over s - y})\). Then the following conditions are equivalent: 1. \(F\) is a polynomial automorphism of \(k^2\), 2. \(J,D \in k \backslash \{0\}\). Hence, in the case of \(p = 0\), we obtain the following equivalent form of the Jacobian conjecture: If \(J \in k \backslash \{0\}\), then \(D \in k \backslash \{0\}\).