The dimension-
Hessian conjecture, denoted 
, asserts that the formal Legendre
transformation of a polynomial in 
variables over a field of characteristic
zero with nonzero constant Hessian
determinant is also a polynomial. More explicitly,
let 
,
where 
denotes the gradient map, and let 
be its local formal inverse, meaning its inverse under
composition as a formal formal power series.
The formal Legendre transform of 
is
 |
(1)
|
and the conjecture states that 
is a polynomial (Meng 2006).
The dimension-
Jacobian conjecture, denoted 
, implies 
, while 
implies 
. Consequently, their all-dimensional forms are equivalent.
De Bondt (2015) proved 
for 
.
Meng and Yang (2026) gave an explicit counterexample to 
. In the five variables 
, 
, 
, 
, 
, let 
and
 |  |  |
(2)
|
 |  |  |
(3)
|
Then the degree-14 polynomial
 |
(4)
|
has a constant Hessian determinant and coincident gradient values at two distinct points:
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
Its gradient map is therefore not an injection, so its formal Legendre transform is not a
polynomial. This refutes 
. Adding an independent quadratic variable propagates the
counterexample to every dimension 
.
It follows that the Hessian conjecture is true for 
, false for 
, and open only for 
. Across the Jacobian
and Hessian conjecture families, the only unresolved cases are 
and 
. If the Hessian conjecture is true in dimension 4, then
the Jacobian conjecture is true in dimension 2.
