The asymptotic Gotsman-Linial conjecture on average sensitivity of polynomial threshold functions

Theorem 1.1: for $n\ge1$, $1\le d\le n$ and a real multilinear polynomial $p$ of degree at most $d$, $f=\mathrm{sgn}(p)$ with $\mathrm{sgn}(0)=1$ has $I(f)\le8d\sqrt n$; the constant is absolute and $d$ may grow with $n$. Symmetric examples show the order $d\sqrt n$ is sharp for $d\le\sqrt n$. It removes the polylogarithmic loss in Kane's bound. It makes no claim about the exact extremizers of the original Gotsman-Linial proposal (already refuted by Chapman and by Kim-Maldonado-Wellens) and does not address non-uniform distributions.

Theorem 1.1: for $n\ge1$, $1\le d\le n$ and a real multilinear polynomial $p$ of degree at most $d$, $f=\mathrm{sgn}(p)$ with $\mathrm{sgn}(0)=1$ has $I(f)\le8d\sqrt n$; the constant is absolute and $d$ may grow with $n$. Symmetric examples show the order $d\sqrt n$ is sharp for $d\le\sqrt n$. It removes the polylogarithmic loss in Kane's bound. It makes no claim about the exact extremizers of the original Gotsman-Linial proposal (already refuted by Chapman and by Kim-Maldonado-Wellens) and does not address non-uniform distributions.