关于对称拟凸多项式的不相连定理

Unlinking Theorem for Symmetric Quasi-Convex Polynomials

假定μ_(n)为R^(n)上的标准高斯测度,X为R^(n)上的随机向量,分布为μ_(n).不相连猜测说的是:如果f与g为R^(n)上的两个多项式,而且f(X)与g(X)相互独立,则存在R^(n)上的正交变换Y=LX及整数k使得f◦L^(-1)为(y_(1),y_(2),…,y_(k))的函数,g◦L^(-1)为(y_(k+1),y_(k+2),…,y_(n))的函数.此时,称f与g不相连.在这篇注记中,我们证明:对于两个对称拟凸多项式f与g,如果f(X)与g(X)相互独立,则f与g不相连.

Let μ_(n) be the standard Gaussian measure on R^(n) and X be a random vector on R^(n) with the law μ_(n).U-conjecture states that if f and g are two polynomials on R^(n) such that f(X)and g(X)are independent,then there exist an orthogonal transformation Y=LX on R^(n) and an integer k such that f◦L^(-1) is a function of(y_(1),y_(2),…,y_(k))and g◦L^(-1) is a function of(y_(k+1),y_(k+2),…,y_(n)).In this case,f and g are said to be unlinked.In this note,we prove that two symmetric,quasiconvex polynomials f and g are unlinked if f(X)and g(X)are independent.