Gotsman–Linial conjecture (total influence vs. sign degree)

Prove that for every Boolean function f: {0,1}^n → {0,1}, the total influence Inf[f] is at most O(√n · sdeg(f)), where sdeg(f) denotes the sign degree of f.

Background

The total influence Inf[f] summarizes average sensitivity of a Boolean function, while sign degree sdeg(f) measures the minimum degree of a polynomial that sign-represents f.

The classical Gotsman–Linial conjecture proposes a fundamental upper bound connecting these two measures. The authors restate this long-standing conjecture in the context of their work relating rational degree, sign degree, and decision-tree complexity.