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

Abstract:

Differential equations of the form $r p(z)q’(z) - sp’(z)q(z) = \gamma p(z)$ or $rp(z)q’(z) - sp’(z)q(z) = \gamma$, where $\gamma$ 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 $\mathbb {C}[z]$, there are only finitely many extra-special pairs of a given degree $n$. This implies that, modulo automorphisms of $\mathbb {C}[z]$, 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 $s$.