AI 中文总结
本文证明了四维情形下的四次Hessian猜想，将其归约为特定形式的多项式，结合三维、二维Hessian猜想及秩分析完成论证。
AI 中文摘要
Hessian猜想询问：具有非零常数Hessian行列式的多项式是否存在多项式梯度逆。已知该猜想在维数不超过3时成立，在维数至少为5时不成立，在四维时仍未解决。本文证明了其四变量四次情形。该多项式的最高次齐次部分的Hessian行列式为零，根据四维齐次Hesse定理，它是一个锥。我们将其锥代表划分为三类：具有非零三元Hessian的真正三元四次多项式、真正二元四次多项式，以及一个线性形式的四次幂。在第一类中，七次行列式方程迫使三次部分在锥方向上为仿射线性；在二元类型中，六次方程在三次部分的二元Hessian中给出一个零常方向；在一元类型中，五次方程和一个常方向引理给出相同结论。因此，每一类都可简化为\\[ f=P(x_1,x_2,x_3)+x_4Q(x_1,x_2,x_3)+a x_4^2, \qquad °Q\leq2. \\]我们独立于\\(P\\)的次数或最高次部分，证明所有这种形式的常Hessian多项式都具有多项式梯度逆。当\\(a\neq0\\)时，通过Schur补可归约为已知的三维Hessian猜想；当\\(a=0\\)时，各向同性锥秩分析排除了秩为2的情况，通过显式三角逆解决了秩为1的例外情况，并将秩为0的情况归约为二维Hessian猜想。整个过程中保留了耦合的六次恒等式，未分离关于固定二次型的任何分量。
英文摘要
The Hessian conjecture asks whether a polynomial with nonzero constant Hessian determinant has a polynomial gradient inverse. It is known in dimensions at most three, false in dimensions at least five, and open in dimension four. We prove its four-variable quartic case. The top homogeneous part has zero Hessian determinant and, by the four-dimensional homogeneous Hesse theorem, is a cone. We divide its cone representative into three exhaustive types: a genuinely ternary quartic with nonzero ternary Hessian, a genuinely binary quartic, and a fourth power of a linear form. In the first type, the degree-seven determinant equation forces the cubic part to be affine-linear in the cone direction. In the binary type, the degree-six equation gives a constant null direction in a two-variable Hessian of the cubic part. In the unary type, the degree-five equation and a constant-direction lemma give the same conclusion. Every type therefore reduces to \[ f=P(x_1,x_2,x_3)+x_4Q(x_1,x_2,x_3)+a x_4^2, \qquad °Q\leq2. \] We prove, independently of the degree or top part of \(P\), that every constant-Hessian polynomial of this form has a polynomial gradient inverse. The branch \(a\ne0\) descends from the known three-dimensional Hessian conjecture after a Schur complement. When \(a=0\), an isotropic-cone rank analysis eliminates rank two, solves the rank-one exception by an explicit triangular inverse, and reduces rank zero to the two-dimensional Hessian conjecture. The coupled degree-six identity is retained throughout; no component with respect to a fixed quadratic form is separated.
