Jacobian pairs, D-resultants, and automorphisms of the plane Original Research Article

Let F = (f,g): k2 → k2 be a polynomial mapping over a field k, with f,g ε k[x, y]. The principal results are: (1) if F is a polynomial automorphism then J, the Jacobian determinant of F with respect to (x, y), and D, the D-resultant of f and g with respect to y, are both non-zero elements of k; (2) if the characteristic of k is zero, or it does not divide the degree of the extension k(x, y) superset of or equal to k(f, g), then the converse is true as well. For k of characteristic zero, this yields a new, algebraically computable characterization of polynomial automorphisms of k2, and hence a reformulation of the two variable Jacobian conjecture.