Theorems of W. W. Stothers and the Jacobian Conjecture in two variables

Abstract

Differential equations of the form rp(z)q'(z) ― sp'(z)q(z) = γp(z) or rp(z)q'(z) ― sp'(z)q(z) = ―, where ― is a nonzero complex number and r, s are positive integers, have arisen in attempts to solve the two-variable Jacobian Conjecture. Solutions of such equations, in which p(z) and q(z) are monic complex polynomials of positive degrees r and s, give rise to extra-special pairs of polynomials in the sense of W. W. Stothers. Stothers showed that, modulo automorphisms of ℂ[z], there are only finitely many extra-special pairs of a given degree n. This implies that, modulo automorphisms of ℂ[z], there are only finitely many solutions of the above differential equations in which p(z) and q(z) are monic polynomials of given degrees r and s.