Average sensitivity of polynomial threshold functions

For every n ≥ 1 and $1\le d\le n$, we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most $8d\sqrt{n}$. This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention $\mathrm{sgn}(0)=1$.