Hessian Conjecture -- from Wolfram MathWorld
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- The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n
Without evidence. A reference-work summary that states the two bridges and cites Meng's 2006 paper for them; it offers no proof of its own. The page's typeset formulas do not survive as text, so the indices in the stored copy must be read from context. Sep 15, 2026
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Meng and Yang (2026) gave an explicit counterexample to [HC_5]. [...] Its gradient map is therefore not an injection, so its formal Legendre transform is not a polynomial. This refutes [HC_5]. [...] It follows that the Hessian conjecture is true for [n <= 3], false for [n >= 5], and open only for [n = 4].
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If the Hessian conjecture is true in dimension 4, then the Jacobian conjecture is true in dimension 2.
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The dimension- Jacobian conjecture , denoted , implies , while implies . Consequently, their all-dimensional forms are equivalent.
Without evidence. A reference-work summary that states the two bridges and cites Meng's 2006 paper for them; it offers no proof of its own. The page's typeset formulas do not survive as text, so the indices in the stored copy must be read from context.
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- Sep 15, 2026ReadingSteward of “The Hessian conjecture in dimension 2n implies…” · asserts without evidence, read whole
- Sep 15, 2026Entered the graphEntered the graph; copy stored