对称拟凸多项式的解链定理
Unlinking Theorem for Symmetric Quasi-convex Polynomials
摘要：
令$μ_n$为$\mathbb{R}^n$上的标准高斯测度，$X$为$\mathbb{R}^n$上的随机向量，其规律为$μ_n$。U猜想指出，如果$f$和$g$是$\mathbb{R}^n$上的两个多项式，使得$f(X)$和$g(X)$是独立的，则存在$\mathbb{R}^n$上的正交变换$L$和整数$k$，使得$f\circ L$是$(x_1,\cdots,x_k)$的函数，$g\circ L$是$(x_{k+1},\cdots,x_n)$的函数。在这种情况下，$f$和$g$被称为未链接。本文证明了当$f(X)$和$g(X)$相互独立时，两个对称的拟凸多项式$f$和$g$是不连通的.
英文摘要：
Let $μ_n$ be the standard Gaussian measure on $\mathbb{R}^n$ and $X$ be a random vector on $\mathbb{R}^n$ with the law $μ_n$. U-conjecture states that if $f$ and $g$ are two polynomials on $\mathbb{R}^n$ such that $f(X)$ and $g(X)$ are independent, then there exist an orthogonal transformation $L$ on $\mathbb{R}^n$ and an integer $k$ such that $f\circ L$ is a function of $(x_1,\cdots,x_k)$ and $g\circ L$ is a function of $(x_{k+1},\cdots,x_n)$. In this case, $f$ and $g$ are said to be unlinked. In this note, we prove that two symmetric, quasi-convex polynomials $f$ and $g$ are unlinked if $f(X)$ and $g(X)$ are independent.
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