AI 中文总结
本文给出一个五变量整数多项式\(\Psi\)作为黑塞猜想\(\HC_5\)的反例，通过单变量舒尔下降从阿尔波格的六变量雅可比反例翻倍得到。结合相关定理及反驳，确定了除\(n = 4\)外各维度黑塞猜想的真假，还给出六变量\(\HC_6\)反例，扩展并取代了作者早期笔记。
AI 中文摘要
我们展示了一个明确的五变量整数多项式\(\Psi\)，其总次数为\(14\)且黑塞行列式\(\det\Hess\Psi\equiv128\)恒定，但其梯度不是单射的。因此，\(\Psi\)的形式勒让德变换不是多项式，黑塞猜想\(\HC_5\)不成立。该反例是通过单变量的舒尔下降（单个变量的部分勒让德变换）从阿尔波格的六变量雅可比反例翻倍得到的，其所有定义恒等式都已在精确有理算术下进行了检验。结合德·邦特关于\(n\leq3\)时\(\HC_n\)成立的定理，以及将雅可比猜想\(\JC_n\)与黑塞猜想\(\HC_n\)联系起来的基本翻倍和稳定化桥梁，再加上阿尔波格对\(\JC_3\)的反驳，这就确定了除\(n = 4\)外每个维度的黑塞猜想：\(n\leq3\)时\(\HC_n\)为真，\(n\geq5\)时为假，仅在\(n = 4\)时未解决。两个家族中恰好有两个陈述未解决，即\(\JC_2\)和\(\HC_4\)，由\(\HC_4\Rightarrow\JC_2\)联系起来。在此过程中，我们还记录了一个明确的六变量\(\HC_6\)反例，其黑塞行列式恒定为\(-4\)，在\(\C\)上梯度一般为三对一。本文扩展并取代了第一作者早期的教育笔记。
英文摘要
We exhibit an explicit integer polynomial in five variables, of total degree $14$ and with constant Hessian determinant $128$, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture $\HC_5$ is false. The counterexample is obtained from the six-variable doubling of Alpöge's 2026 Jacobian counterexample by a one-variable \emph{Schur descent}---a partial Legendre transform in a single variable. Combined with de~Bondt's theorem that $\HC_n$ holds for $n\le3$, with the elementary doubling and stabilization bridges relating the Jacobian conjectures $\JC_n$ to the Hessian conjectures $\HC_n$, and with Alpöge's refutation of $\JC_3$, this decides the Hessian conjecture in every dimension except $n=4$: $\HC_n$ is true for $n\le3$, false for $n\ge5$, and open only at $n=4$. Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$. Along the way we record, as a warm-up, an explicit six-variable counterexample to $\HC_6$ with constant Hessian determinant $-4$ and non-injective gradient. This note adds the five-variable counterexample to, and updates the status recorded in, the first author's earlier educational preprint \cite{MengRG2026}.
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