Unlinking Theorem for Symmetric Quasi-convex Polynomials

Let ${\mu }_n$ be the standard Gaussian measure on ${\mathbb{R}}^n$ and $X$ be a random vector on ${\mathbb{R}}^n$ with the law ${\mu }_n$. 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.