Unlinking Theorem for Symmetric Quasi-convex Polynomials

TL;DR

This paper proves that symmetric, quasi-convex polynomials are unlinked if their evaluations on a Gaussian vector are independent, confirming a special case of the U-conjecture in probability theory.

Contribution

It establishes that symmetric, quasi-convex polynomials with independent Gaussian evaluations are necessarily unlinked, advancing understanding of polynomial independence under Gaussian measures.

Findings

Proves symmetric, quasi-convex polynomials are unlinked if independent.

Supports the U-conjecture for a specific class of polynomials.

Clarifies the structure of independent polynomial evaluations in Gaussian space.

Abstract

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.