The gotsman-linial conjecture is false

Abstract

In 1991, Craig Gotsman and Nathan Linial conjectured that for all n and d, the average sensitivity of a degree-d polynomial threshold function on n variables is maximized by the degree-d symmetric polynomial which computes the parity function on the d layers of the hypercube with Hamming weight closest to n=2. We refute the conjecture for almost all d and for almost all n, and we confirm the conjecture in many of the remaining cases.