The Gotsman-Linial Conjecture is False

TL;DR

This paper disproves the Gotsman-Linial conjecture by showing that the proposed symmetric polynomial does not always maximize average sensitivity of degree-d polynomial threshold functions, for most values of n and d.

Contribution

The paper provides a counterexample to the Gotsman-Linial conjecture and clarifies the conditions under which the conjecture does or does not hold.

Findings

The conjecture is false for almost all d and n.

The conjecture holds in many remaining cases.

Counterexamples demonstrate the limits of the conjecture's validity.

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.