On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems

Abstract: We study the 4D Hessian conjecture in Lorentzian signature. For a polynomial potential $φ$ in 4 real variables whose Hessian matrix has inertia index 1 and determinant $-1$, we define a pivot of $φ$ as a direction vector $v$ such that the double derivative $D<sup>2_vφ$ is a constant function. We then prove that the gradient mapping of every potential admitting a pivot is a regular automorphism, and that a pivot always exists when $φ$ decomposes into homogeneous pieces as $φ=φ<em>d+φ</em>{d-1}+φ<em>2+φ_1+φ_0$ with $d\geq4$. More generally, we prove the same conclusion when $φ=φ_d+\cdots+φ</em>{d-k}+φ<em>2+φ_1+φ_0,$ where $k\geq0$ and $d\geq4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism $μ</em>φ$. We prove that the existence of a pivot is equivalent to $\operatorname{rank}(μ_φ)\leq55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that a Hesse system consisting entirely of singular matrices has complex dimension at most 6, and equality forces a pivot.

• The paper uncovers a key result with unimodular Hessian potentials of index 1, showcasing a polynomial-automorphic gradient map through the usage of constant pivots.
• Demonstrates three mechanisms for creating constant pivots, covering highest-weight/Newton-polytope techniques, degree separation for layered potentials, and a complex linear-algebraic approach.
• Determines whether a polynomial potential with a constant pivot implies a polynomial inverse, building on differential geometric and algebraic constraints
• follow_up_questions

Setting and motivation

The Hessian conjecture, in its modern formulation due to Meng [Meng2006], asks whether a polynomial potential $\phi$ whose Hessian determinant is a nonzero constant has a gradient mapping $\nabla\phi$ that is a polynomial automorphism. It sits on the symmetric-gradient branch of Keller's Jacobian conjecture: the Jacobian conjecture implies the Hessian conjecture dimension-by-dimension, while the Hessian conjecture in dimension $2n$ implies the Jacobian conjecture in dimension $n$ via the doubled potential $\langle F(x),y\rangle$ of a Keller mapping. The reductions of Bass–Connell–Wright and de Bondt–van den Essen reduce the all-dimensional problem to nilpotent Hessians of homogeneous quartics, which makes constant-Hessian potentials a central test case.

The paper under review, by Hanwen Liu (2608.19112), studies the four-dimensional real case with Hessian index 1 — Lorentzian signature. This is precisely the remaining open case: positive-definite Hessians are settled over $\mathbb{R}$ by Meng using the Jörgens–Calabi–Pogorelov rigidity theorem for the constant Monge–Ampère equation; dimensions two and three were settled by Dillen and de Bondt respectively; and, following the 2026 five-variable counterexample of Meng–Yang built from Alpöge's counterexample to the Jacobian conjecture (degree 14, Hessian determinant 128, non-injective gradient), dimensions at least five are false in both conjectures. Thus the four-dimensional Lorentzian case includes the quartic case equivalent to the planar Jacobian conjecture.

The paper's central device is elementary to state: a constant pivot for $\phi$ is a nonzero constant vector $v$ such that $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$ is constant. Its main structural result is that every unimodular Hessian potential of index 1 admitting a pivot has a polynomial-automorphic gradient map, and it develops three independent mechanisms for producing pivots: highest-weight/Newton-polytope methods, degree separation for layered potentials, and a complex linear-algebraic criterion via an associated system of quadrics.

Constant pivots imply polynomial inversion

The inversion argument is organized by the causal type of the pivot relative to the Lorentzian form. If $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$, a timelike analysis forces $\nabla\phi$0 itself to be quadratic: writing $\nabla\phi$1 with $\nabla\phi$2, the pencil of Schur complements $\nabla\phi$3 consists of strictly convex polynomials with Hessian determinant $\nabla\phi$4, so Jörgens–Calabi–Pogorelov makes each $\nabla\phi$5 quadratic, and varying $\nabla\phi$6 forces $\nabla\phi$7 affine and $\nabla\phi$8 quadratic.

For a lightlike pivot ($\nabla\phi$9), the potential admits an explicit normal form

$2n$0

with $2n$1 positive definite and $2n$2. The proof uses that level surfaces of $2n$3 have vanishing second fundamental form, hence are parallel affine planes. Crucially, since $2n$4, the inverse matrix $2n$5 equals the adjugate and is polynomial, so explicit recovery of $2n$6 from $2n$7 yields a polynomial inverse. For a spacelike pivot ($2n$8), the Schur complement $2n$9 has Hessian determinant $n$0 over the rational function field $n$1; de Bondt's three-dimensional theorem gives a rational inverse, and the birational Keller theorem upgrades this to a polynomial automorphism of $n$2, hence of the original gradient map. Together these establish the Constant-Pivot Theorem: existence of a pivot implies $n$3 is a polynomial automorphism.

Two auxiliary facts used repeatedly are worth noting. First, if a flat Lorentzian pencil $n$4 preserves $n$5 with $n$6 of signature $n$7, then $n$8 — a purely signature-theoretic constraint obtained by eigenvalue-counting along rays. Second, a version of the small-rank Hessian theorem: a homogeneous quartic-or-higher form in four variables with generic Hessian rank at most 2 depends on at most two linear forms, with a constant kernel of dimension at least 2.

Pivot existence for two-layer potentials

The first main existence theorem covers potentials of degree $n$9 of the two-layer form $\langle F(x),y\rangle$0. The mechanism is dilation separation: substituting $\langle F(x),y\rangle$1 scales the coefficient of $\langle F(x),y\rangle$2 monomially, and for $\langle F(x),y\rangle$3 the exponent map $\langle F(x),y\rangle$4 is injective on the simplex $\langle F(x),y\rangle$5, forcing all mixed determinant coefficients to vanish. The borderline case $\langle F(x),y\rangle$6 requires essential rank $\langle F(x),y\rangle$7 of $\langle F(x),y\rangle$8, where the unique collision between the $\langle F(x),y\rangle$9 and $\mathbb{R}$0 coefficients is defused because $\mathbb{R}$1 kills the $\mathbb{R}$2 coefficient.

From the resulting flat-pencil identity, Lemma lorentz_flat_pencil gives generic ranks at most 2 for both top layers; kernel intersection counting then produces $\mathbb{R}$3 annihilated by both $\mathbb{R}$4 and $\mathbb{R}$5 pointwise, making $\mathbb{R}$6 constant. The exceptional case $\mathbb{R}$7, $\mathbb{R}$8 is handled directly: the $\mathbb{R}$9 coefficient factors as $\phi$0, forcing the second factor to vanish identically while $\phi$1 spans the common kernel of $\phi$2.

The quintic case splits by essential rank. When $\phi$3, a dichotomy lemma for positive semi-definite singular $\phi$4 matrices of quadratic forms either yields a common kernel directly or forces the degenerate structure $\phi$5; the latter alternative is eliminated by a weighted highest-degree computation producing a weight-$\phi$6 leading part $\phi$7 with $\phi$8, contradicting constancy of the full determinant via the weight formula. When $\phi$9, asymptotics of the transverse Schur complement show the transverse quartic Hessian $v$0 is everywhere positive semi-definite and singular; a convexity lemma (proved via invariance of domain plus Sard's theorem) supplies a direction in its kernel, giving a pivot. The quartic case proceeds by essential rank through a singular binary pencil lemma for linear systems of singular symmetric $v$1 matrices.

Assembling these cases proves the Two-Layer Pivot Theorem, and combined with the inversion results this settles the Hessian conjecture for all index-1 four-dimensional potentials of the two-layer form.

The Hesse system and the rank criterion

The second half introduces a degree-free algebraic object. Writing $v$2, the Hesse system $v$3 is the span of all differences $v$4 as quadratic forms in $v$5; a vector is a complex constant pivot exactly when all quadrics in $v$6 vanish at it. With the multiplication map $v$7, the key criterion is:

$v$8

Here 56 is $v$9. Surjectivity when no common zero exists follows from a Hilbert-series computation: four general quadrics without common zero form a regular sequence with quotient Hilbert series $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$0, whose degree-five component vanishes, so $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$1 lies in the image of $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$2. This is a finite, directly computable criterion depending only on the coefficients of $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$3.

A substantial technical contribution is the complex-to-real principle: any complex constant pivot for an index-1 potential forces a real one. Assuming no real pivot exists, the real and imaginary parts of a complex pivot span a screen carrying the normal form $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$4 with $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$5. The proof combines the Hartman–Nirenberg cylinder theorem applied to the scalar profile $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$6 (or $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$7 when $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$8 is scalar), comparison of $$D_v^2\phi = \langle \operatorname{Hess}(\phi)v,v\rangle$$9-degree components in the Schur-complement identity, and finally a one-parameter analysis showing that either the kernel of the $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$0 matrix family $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$1 is constant or $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$2 is affine in $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$3 — both yielding a real pivot.

Combining the two propositions gives the CR Pivot Criterion: for a unimodular index-1 potential, a constant pivot exists if and only if $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$4. Consequently, whenever this finite check succeeds, the gradient mapping is a polynomial automorphism.

Applications to low-dimensional and singular Hesse systems

Two geometric applications convert the criterion into verifiable hypotheses. If $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$5, projective dimension theory guarantees a common zero of the quadrics, hence a pivot. For the borderline dimension 4, the argument is more delicate: assuming no pivot, the quadrics have empty base locus, and a dominance lemma shows the Hessian-difference map $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$6 must be Zariski-dominant (the proof controls the rank stratification of $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$7 against that of the quadric evaluation map $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$8, which has finite fibres). Dominance forces $\langle \operatorname{Hess}(\phi)v,v\rangle < 0$9 to vanish on all of $\nabla\phi$00, making each $\nabla\phi$01 nilpotent and self-adjoint for the Lorentzian form $\nabla\phi$02. But the trace pairing on self-adjoint endomorphisms of signature $\nabla\phi$03 has signature $\nabla\phi$04, so a totally isotropic subspace of nilpotents has real dimension at most 3 — contradicting $\nabla\phi$05. Hence:

If the Hesse system has complex dimension at most 4, the Hessian conjecture holds for $\nabla\phi$06.

Second, if every member of $\nabla\phi$07 is a singular matrix, a classification of linear subspaces of $\nabla\phi$08 consisting of singular matrices gives $\nabla\phi$09, with equality forcing the block-diagonal model $\nabla\phi$10, whose common kernel provides a complex pivot and hence a real one. In fact Bertini's theorem improves this substantially: a projective linear system of quadrics with empty base locus contains a smooth member, so a fully singular Hesse system always has a base point, and therefore every such potential — regardless of dimension of $\nabla\phi$11 — admits a constant pivot.

Finally, the multi-layer analysis extends the two-layer theorem. Using the Cauchy–Binet/Sylvester identity for ridge perturbations $\nabla\phi$12 — a representation which the paper observes generates all solutions of the Monge–Ampère equation up to affine terms — and a flat-pencil rank-collapse lemma reducing to de Bondt's small-rank classification, the paper proves:

If $\nabla\phi$13 with $\nabla\phi$14, then a pivot exists, and more generally whenever the degree set $\nabla\phi$15 is 4-separated in the sense that the Minkowski sums $\nabla\phi$16 are pairwise disjoint. The interval-disjointness argument forces the coefficients $\nabla\phi$17 of $\nabla\phi$18 to vanish individually, and rank collapse then produces a common kernel direction. Note the growth condition $\nabla\phi$19: the method genuinely fails when many high layers crowd together, e.g. dense potentials with degrees $\nabla\phi$20 present.

Limitations and open questions

The results leave the general four-dimensional Lorentzian Hessian conjecture unresolved. Concretely: (i) the multi-layer theorem requires $\nabla\phi$21, so potentials with more than one nonlinear layer below the threshold — including arbitrary dense polynomials — are not covered by the degree-separation route, though they remain accessible in principle via the rank-55 criterion if $\nabla\phi$22 can be bounded; (ii) no upper bound on $\nabla\phi$23 is established for general potentials, so the dimension-4 application does not exhaust the quartic case relevant to the planar Jacobian conjecture; (iii) the complex-to-real principle relies essentially on index 1 and does not extend verbatim to higher indefinite signatures; (iv) the quintic elimination argument depends on the specific weight assignment and does not obviously generalize. Whether the Hesse-system framework can be pushed to cover all quartic index-1 potentials remains the sharpest open question raised by the paper.

Conclusion

This paper establishes the Hessian conjecture for a broad class of four-dimensional index-1 potentials organized around a single combinatorial device, the constant pivot. Three contributions stand out: the causal trichotomy reducing pivots to known low-dimensional theorems; the equivalence between pivot existence and the finite rank bound $\nabla\phi$24, made real by the complex-to-real principle; and the geometric corollaries settling all potentials with Hesse dimension at most 4 or with entirely singular Hesse systems. The framework converts a PDE-flavored rigidity question into computable finite-dimensional linear algebra over $\nabla\phi$25, providing a concrete template for attacking the remaining quartic case.