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Separable Jacobian Conjecture

The separable Jacobian conjecture asserts that a polynomial map F=(f_1,...,f_n):k^n->k^n over an algebraically closed field k of positive field characteristic p, with nonzero constant Jacobian determinant and finite function field degree [k(x_1,...,x_n):k(f_1,...,f_n)] relatively prime to p, is an automorphism.

Mondello (2026) refuted this statement in dimension 2 over the algebraic closure k of F_2. The polynomial map F=(P,Q) with

P(x,y)=x+x^2y+x^4+x^6y^2
(1)
Q(x,y)=y+x^5+x^6y+x^7y^2+x^8y^3
(2)

has Jacobian determinant 1, but

F(0,1)=F(1,0)=F(1,1)=(0,1).
(3)

Moreover, k(x,y)/k(P,Q) is a separable extension of degree 3. The plane Jacobian conjecture over the complex numbers remains open.

See also

Field Characteristic, Jacobian Conjecture, Polynomial Map, Separable Extension

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References

Mondello, R. "A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two." 29 Jul 2026. https://arxiv.org/abs/2608.02634.

Cite this as:

Weisstein, Eric W. "Separable Jacobian Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SeparableJacobianConjecture.html

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