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

TL;DR

This paper proposes a new set of equations in characteristic p that potentially fully characterize invertible polynomial maps, offering an alternative formulation of the Jacobian Conjecture in positive characteristic.

Contribution

It introduces a novel formulation of the Jacobian Conjecture in characteristic p, constructing equations that may fully determine invertibility of polynomial maps in this setting.

Findings

Constructed equations that potentially characterize invertibility in characteristic p.

Linked the new formulation to the classical Jacobian Conjecture in characteristic zero.

Explored special cases to strengthen the new formulation.

Abstract

The Jacobian Conjecture uses the equation $det(Jac(F))\in k^*$, 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.