On the Jacobian conjecture in characteristic zero

Abstract

We study the Jacobian conjecture for Keller maps $f:X_0:=\mathbf{A}^n\rightarrow Y_0:=\mathbf{A}^n$ in characteristic $0$ and attempt to prove it. We are quite aware of the fact that many people have tried to prove the Jacobian conjecture before us and hence we stress that this manuscripts is only an attempt. Our approach is to study the finiteness variety $V_f \subset Y_0$ of $f$, the set of points of $Y_0$ over which $f$ fails to be proper. We study a general component $V \subset V_f$ of this set by introducing a suitable representation of $X_0$ which we call the $u-\gamma$ representation. This view of $X_0$ has the advantage that it allows us to explicitly write down the Jacobian matrix of $f$. We then turn our attention to the condition that $|J(f)| = 1$, which we interpret as a partial differential equation in one unknown function. We study the characteristics of this equation and prove that the dynamics of this is strongly related to the ramification $K$ above $V$. We then study the action of a certain cyclic Galois group induced by the $u-\gamma$ action on these differential equations and prove that the growth of the functions are bounded. Alternatively we prove the same result via a vector field argument where we deduce that if $K > 0$ then the function $u^{-K}$ is in fact an analytic function around $v \in V \subset Y_0$. This leads to $K = 1$ and as $\pi_1(Y_0 - S) \simeq \pi_1(Y_0)$ if $S$ is of codimension at least two, the Jacobian conjecture follows immediately.