When the Jacobian Lies: Non-Invertible Maps in Positive Characteristic
This presentation explores the surprising failure of classical intuition about polynomial maps in positive characteristic. The authors introduce a systematic framework of geometric invariants for étale endomorphisms of affine space, then use it to construct explicit counterexamples to multiple Jacobian conjectures. The talk focuses on how maps with constant Jacobian determinant can have arbitrary image gaps, controlled fiber structures, and exotic associated geometries, revealing fundamental differences between characteristic zero and positive characteristic behavior.Script
In characteristic zero, a polynomial map with constant Jacobian determinant should be invertible. In positive characteristic, that intuition collapses: the authors construct explicit maps that are perfectly smooth everywhere yet fail to hit even a single point.
The culprit is the Frobenius system, a polynomial equation of the form x plus a function of x to the p equals zero. The paper disproves a longstanding conjecture by exhibiting such systems with no solutions at all over algebraically closed fields.
Theorem 1 delivers precise numerical control. For any geometric degree p times m and any nonnegative integer s, there exists an étale endomorphism of the affine plane whose image omits exactly s points and whose fiber cardinalities range from zero to p times m.
Each étale endomorphism defines a normalization called the associated Jacobian variety. In dimension two, the boundary consists of rational curves isomorphic to the affine line, and there is at most one singular point, which the paper proves must be a quotient singularity.
When the image complement is finite, iteration eventually stabilizes in dimensions one and two. But in dimension three and higher, the paper constructs endomorphisms whose images shrink at every iterate, never reaching equilibrium despite starting with a finite gap.
The moduli space of Jacobian maps is connected but almost always singular, smooth only when the degree or dimension is one, or in the exceptional case where both equal two in characteristic two. To explore these geometric mysteries further and create your own mathematical videos, visit EmergentMind.com.
