The Legendre potential and the gradient case of the differential Galois criterion for the Jacobian Conjecture
Abstract
Let h∈C[X1,…,Xn] have constant nonzero Hessian determinant, let h* be its formal Legendre transform and g=vvX,grad h−h. The Nambu derivations of grad h satisfy δi(g)=Xi and δα(g)=(∂αh*)∘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∈C[∂1h,…,∂nh]. Moreover, grad h is a polynomial automorphism if and only if δα(g)=0 for |α|=(°h−1)n−1+2, and then the coefficients δα(g)(x0)/α! of h* at grad h(x0) 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(∂1h,…,∂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 JC2 and HC4.
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BibTeX
@misc{adamus2026legendre,
title = {{The Legendre potential and the gradient case of the differential Galois criterion for the Jacobian Conjecture}},
author = {Elżbieta Adamus and Zbigniew Hajto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36736},
eprint = {2610.36736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36736},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Legendre potential and the gradient case of the differential Galois criterion for the Jacobian Conjecture
AU - Elżbieta Adamus
AU - Zbigniew Hajto
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36736
UR - https://arxiv.org/abs/2610.36736
AB - Let h∈C[X_1,…,X_n] have constant nonzero Hessian determinant, let h^* be its formal Legendre transform and g=vvX,grad h−h. The Nambu derivations of grad h satisfy δ_i(g)=X_i and δ^α(g)=(∂^αh^*)∘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∈C[∂_1h,…,∂_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(∂_1h,…,∂_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.
ER -