  In this paper, we consider low-degree polynomials of inner products between a
collection of random vectors. We give an almost orthogonal basis for this
vector space of polynomials when the random vectors are Gaussian, spherical, or
Boolean. In all three cases, our basis admits an interesting combinatorial
description based on the topology of the underlying graph of inner products.
  We also analyze the expected value of the product of two polynomials in our
basis. In all three cases, we show that this expected value can be expressed in
terms of collections of matchings on the underlying graph of inner products. In
the Gaussian and Boolean cases, we show that this expected value is always
non-negative. In the spherical case, we show that this expected value can be
negative but we conjecture that if the underlying graph of inner products is
planar then this expected value will always be non-negative.
