Gaussian variables, polynomials and permanents Original Research Article

We prove that given n unit vectors (aj)nj=1 in a complex Hilbert space there exists a unit vector x such that imageleft angle bracketx,a1right-pointing angle bracket…left angle bracketx,anright-pointing angle bracketnot greater-than-or-equaln−n/2. This answers a question about the best constant in an inequality for polynomials in several variables. The proof depends on a relation between gaussian variables and permanents.