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

We study the 4D Hessian conjecture in Lorentzian signature. For a polynomial potential $\phi$ in 4 real variables whose Hessian matrix has inertia index 1 and determinant $-1$, we define a pivot of $\phi$ as a direction vector $v$ such that the double derivative $D_v^2\phi$ 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 $\phi$ decomposes into homogeneous pieces as $\phi ={\phi }_d+{\phi }_{d-1}+{\phi }_2+{\phi }_1+{\phi }_0$ with $d\ge 4$. More generally, we prove the same conclusion when $$\phi ={\phi }_d+\cdots +{\phi }_{d-k}+{\phi }_2+{\phi }_1+{\phi }_0,$$ where $k\ge 0$ and $d\ge 4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism ${\mu }_{\phi }$. We prove that the existence of a pivot is equivalent to $\mathrm{rank}({\mu }_{\phi })\le 55$. 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.