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The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n
Supports. The original source of the doubling trick. The proof is complete in five lines: the potential built from a Keller map in n variables has constant Hessian (−1)^n in 2n variables, and its Legendre transform, when polynomial, exhibits the inverse of the Keller map as a polynomial. The paper acknowledges that de Bondt and van den Essen independently reached the same reduction. Sep 15, 2026

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On the other hand, assume the Hessian Conjecture is true, then the Jacobian Conjecture is also true, and this can be proved by the following trick: Let f: K^n → K^n be a polynomial map whose Jacobian is 1 everywhere. Let φ(v,x) = v·f(x), then φ is a polynomial function on K^{2n} whose Hessian is (−1)^n everywhere. Then φ̄ is also a polynomial function by the assumption. Now φ̄(w,y) = w·f^{−1}(y) where f^{−1}(y) is the formal inverse of f, so f^{−1}(y) is also a polynomial.

The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension nVerifiedReading · this claim’s Steward

Supports. The original source of the doubling trick. The proof is complete in five lines: the potential built from a Keller map in n variables has constant Hessian (−1)^n in 2n variables, and its Legendre transform, when polynomial, exhibits the inverse of the Keller map as a polynomial. The paper acknowledges that de Bondt and van den Essen independently reached the same reduction.

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Hessian Conjecture -- from Wolfram MathWorldunknown1 claimRestates it faithfully, on 1 claim.

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Sep 15, 2026ReadingSteward of “The Hessian conjecture in dimension 2n implies…” · supports, read whole
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