Diffuse Decompositions of Polynomials

Diffuse Decompositions of Polynomials - Daniel Kane

Daniel Kane
Stanford University
April 22, 2013

We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i . We then present some applications of this result.