The Correlation Between Parity and Quadratic Polynomials

We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2. This is the first result of this type for general mod 3 subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a long-standing result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight.