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

Stefan Maubach ¹   ; Abdul Rauf ²

¹ Radboud University Nijmegen Postbus 9010 6500 GL Nijmegen, The Netherlands
² Department of Mathematics Jacobs University Bremen Bremen, Germany

Colloquium Mathematicum, Tome 146 (2017) no. 1, pp. 15-30

Résumé

The Jacobian Conjecture uses the equation $\mathop {\rm det}\mathop {\rm Jac}(F)\in k^*$, 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. We describe how to construct the conjecturally sufficient equations in characteristic $p$ forcing a polynomial map to be invertible. This provides a formulation of the Jacobian Conjecture in characteristic $p$, alternative to Adjamagbo’s. We strengthen this formulation by investigating some special cases and by linking it to the regular Jacobian Conjecture in characteristic zero.

Détail

DOI : 10.4064/cm6692-3-2016

Keywords: jacobian conjecture uses equation mathop det mathop jac * which short write down many equations putting restrictions coefficients polynomial map characteristic these equations suffice conjecturally force polynomial map invertible describe construct conjecturally sufficient equations characteristic forcing polynomial map invertible provides formulation jacobian conjecture characteristic alternative adjamagbo strengthen formulation investigating special cases linking regular jacobian conjecture characteristic zero