Gotsman–Linial Conjecture

Prove that among degree-k polynomial threshold functions f: {−1,1}^n → {−1,1}, the function of the form f(x) = sgn(p(x1 + ⋯ + xn)), where p is a degree-k univariate polynomial that alternates sign on the k+1 values of x1 + ⋯ + xn closest to 0, has maximal total influence.

Background

The conjecture posits that the extremal degree-k PTF for total influence is symmetric and depends only on the sum of inputs. Weaker versions assert O(k)√n or Ok(1)√n upper bounds on total influence for all degree-k PTFs and are related to noise sensitivity bounds.

The case k=2 remains a key milestone, and the Gaussian analogue has been resolved. Current upper bounds on total influence for degree-k PTFs are far from the conjectured tight behavior.