The Divergence-Free Jacobian Conjecture in Dimension Two

Abstract

A special case, called the divergence-free case, of the Jacobian Conjecture in dimension two is proved. This note outlines an argument for a special case of the Jacobian conjecture in dimension two: Suppose F : C → C is a polynomial so that F (0) = 0, F ′(0) = I, det(F ′(z)) = 1, z ∈ C. (1) where I is the identity transformation on C. Write F (x, y) = ( r(x, y) + x s(x, y) + y ) , (x, y) ∈ C where r, s have no non-zero constant or linear terms and observe that detF ′ = {r, s}+∇ · ( r s ) + 1