  We prove a new bound on the average sensitivity of polynomial threshold
functions. In particular we show that a polynomial threshold function of degree
$d$ in at most $n$ variables has average sensitivity at most
$\sqrt{n}(\log(n))^{O(d\log(d))}2^{O(d^2\log(d)}$. For fixed $d$ the exponent
in terms of $n$ in this bound is known to be optimal. This bound makes
significant progress towards the Gotsman-Linial Conjecture which would put the
correct bound at $\Theta(d\sqrt{n})$.
