A new formulation of the Jacobian Conjecture in characteristic $p$
Volume 146 / 2017
Colloquium Mathematicum 146 (2017), 15-30
MSC: Primary 14R15; Secondary 13F20.
DOI: 10.4064/cm6692-3-2016
Published online: 11 July 2016
Abstract
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.
