洛伦兹符号下的黑塞猜想：常支点与黑塞系统
On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems
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中文总结 AI 辅助
该论文研究四维洛伦兹符号下的黑塞猜想，定义多项式势的支点并证明其存在性条件，关联黑塞系统与典范同态，推导支点存在等价条件及梯度映射为多项式自同构的相关结论。
中文摘要 AI 辅助
我们研究四维洛伦兹符号下的黑塞猜想。对于一个含4个实变量的多项式势φ，当其黑塞矩阵的惯性指数为1、行列式为-1时，我们将φ的支点定义为方向向量v，使得二阶导数D²ᵥφ为常数函数。我们证明：每个存在支点的势的梯度映射都是正则自同构；当φ分解为齐次项φ=φ_d+φ_{d-1}+φ₂+φ₁+φ₀且d≥4时，支点必然存在。更一般地，当φ=φ_d+…+φ_{d-k}+φ₂+φ₁+φ₀（其中k≥0且d≥4k+3）时，上述结论同样成立。随后，我们为每个势关联一个二次线性系统，称为黑塞系统，以及一个典范同态μ_φ，证明支点的存在等价于rank(μ_φ)≤55。作为应用，我们证明当黑塞系统的复维数至多为4时，梯度映射是多项式自同构；我们还证明，完全由奇异矩阵构成的黑塞系统的复维数至多为6，且达到该值时必然存在支点。
英文摘要
As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge--Ampère equation $\det(\operatorname{Hess}(ϕ))=\pm1$ is also a polynomial solution. The general Hessian conjecture is false for $n\geq5$, while in Riemannian signature it follows from the Jörgens--Calabi--Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $ϕ$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $ϕ$ to be a nonzero constant vector $ξ$ such that the second directional derivative $D_ξ^2ϕ$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $ϕ$ decomposes into homogeneous pieces as $ϕ=ϕ_d+ϕ_{d-1}+ϕ_2+ϕ_1+ϕ_0$ with $d\geq4$. More generally, we prove the same conclusion when $$ϕ=ϕ_d+\cdots+ϕ_{d-k}+ϕ_2+ϕ_1+ϕ_0,$$ 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 $μ_ϕ$. 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 if ${\det(\operatorname{Hess}(ϕ-ϕ_2))\equiv0}$, then the potential $ϕ$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\operatorname{Hess}(ϕ-ϕ_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.
发表机构
- Mathematics Institute, University of Warwick（华威大学数学研究所）
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