A new formulation of the Jacobian Conjecture in characteristic $p$

Abstract

The Jacobian Conjecture uses the equation det(Jac(F )) ∈ k * , which is a very short way to write down many equations putting restrictions on the coefficients of a polynomial map F . In characteristic p these equations do not suffice to (conjecturally) force a polynomial map to be invertible. In this article, we describe how to construct the conjecturally sufficient equations in characteristic p forcing a polynomial map to be invertible. This provides an (alternative to Adjamagbo's formulation) definition of the Jacobian Conjecture in characteristic p. We strengthen this formulation by investigating some special cases and by linking it to the regular Jacobian Conjecture in characteristic zero.