Unlinking Theorem for Symmetric Quasi-Convex Polynomials

Let \mu_n be the standard Gaussian measure on \mathbbR^n and X be a random vector on \mathbbR^n with the law \mu_n. U-conjecture states that if f and g are two polynomials on \mathbbR^n such that f(X) and g(X) are independent, then there exist an orthogonal transformation Y=LX on \mathbbR^n and an integer k such that f\circ L^-1 is a function of (y_1,y_2,\cdots,y_k) and g\circ L^-1 is a function of (y_k+1,y_k+2,\cdots,y_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.