高斯矩猜想的小反例

中文摘要 AI 辅助

我们给出了三个独立标准实高斯变量的显式复多项式\(P,Q\)，使得对于每个\(m\geq1\)，有\({\mathbb E}(P^m)=0\)，\({\mathbb E}(QP^m)=m!\neq0\)。在自然复线性坐标下，\(P\)有五项且总次数为\(4\)，这表明在每个维度\(n\geq3\)中高斯矩猜想是错误的。我们还给出了一个四个变量的六项三次多项式反例，它首先被发现，并且已经证明在每个\(n\geq4\)时该猜想不成立。这两个例子都源于相同的系数恒等式。搜索是由Levent Alpöge公开宣布的关于雅可比猜想的显式三维反例所引发的。尽管Derksen、van den Essen和Zhao的主要定理在维度上是全局陈述的，但其证明具有固定维度的内容：\(r\)个变量中的一个不可逆三次齐次凯勒映射会导致\({\mathrm GMC}(2r)\)不成立。跟踪所宣布映射的标准Bass--Connell--Wright约化会得到一个\(79\)变量的保守三次齐次反例，从而导致基于路径的\({\mathrm GMC}(158)\)不成立。该路径在最终的高斯步骤是非构造性的，并且没有给出显式多项式\(P,Q\)。下面维度\(4\)和\(3\)中更小的显式反例并非来自所宣布的雅可比映射。

英文摘要

We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alpöge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.

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