Estimation of certain exponential sums arising in complexity theory

Abstract

It is shown that the correlation on { 0 , 1 } n between parity and a polynomial p ( x 1 , … , x n ) ∈ Z [ X 1 , … , X n ] ( mod q ) , q a fixed odd number and p ( X ) of degree d arbitrary but fixed, is exponentially small in n as n → ∞ . An application to circuit complexity, from where the problem originates, is given. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).