trailThe separable Jacobian conjecture holds: every étale polynomial self-map of affine space in positive characteristic with generic degree prime to the characteristic is an automorphism.
In positive characteristic the Jacobian conjecture is standardly formulated as: étale polynomial self-maps of affine space with generic degree prime to the characteristic are automorphisms.
The naive Jacobian conjecture fails in positive characteristic: the polynomial x − x^p has derivative 1 but is not injective.
A polynomial map in positive characteristic with nonzero Jacobian determinant induces a separable algebraic extension of rational function fields.
Positive-characteristic formulations of the Jacobian conjecture other than Adjamagbo's degree-prime-to-p version have been proposed.
Adjamagbo's separable Jacobian conjecture in positive characteristic implies the classical Jacobian conjecture in characteristic zero.
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The separable Jacobian conjecture holds: every étale polynomial self-map of affine space in positive characteristic with generic degree prime to the characteristic is an automorphism.
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