The Legendre potential and the gradient case of the differential Galois criterion for the Jacobian Conjecture

Let $h\in\C[X_1,\dots,X_n]$ have constant nonzero Hessian determinant, let $h^{*}$ be its formal Legendre transform and $g=\vv{X,\grad h}-h$. The Nambu derivations of $\grad h$ satisfy $δ_i(g)=X_i$ and $δ^α(g)=(\partial^αh^{*})\circ\grad h$. Consequently, in the gradient case the differential Galois criterion of \cite{ACH19}, consisting of $n$ membership conditions, reduces to a single membership condition. Equivalently, $\grad h$ is a polynomial automorphism if and only if $g\in\C[\partial_1h,\dots,\partial_nh]$. Moreover, $\grad h$ is a polynomial automorphism if and only if $δ^α(g)=0$ for $|α|=(°h-1)^{n-1}+2$, and then the coefficients $δ^α(g)(x_0)/α!$ of $h^{*}$ at $\grad h(x_0)$ determine the inverse. If the Hessian determinant of $h$ is merely a nonzero polynomial, then $g$ is still a primitive element of $\C(X)$ over $\C(\partial_1h,\dots,\partial_nh)$. For the Nambu-tower criterion, doubling a Keller map yields no independent conditions. After an affine change of coordinates putting $\grad h$ in the normal form $X+H$, the formal Legendre transform is the sum of the formal Pascal difference series of $g$, and the termination of this series implies polynomial invertibility. In at most three variables the normal form of every gradient Keller map has strongly nilpotent Jacobian part and is therefore Pascal finite, by a theorem of de Bondt, and an explicit gradient automorphism in four variables has a normal form which is not Pascal finite. The five-variable counterexample of Meng and Yang shows that the properties used here do not imply the Hessian conjecture. The results do not settle the open cases $\JC_2$ and $\HC_4$.