Extremal properties of polynomial threshold functions

We give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: 1) Almost every Boolean function has PTF degree at most n/2+O(√(n log n)). Together with results of Anthony and Alon, we establish a conjecture of Wang and Williams [1991] and Aspnes, Beigel, Furst, and Rudich [1994] up to lower order terms. 2) Every Boolean function has PTF density at most (1-1/O(n))2ⁿ. This improves a result of Gotsman [1989]. 3) Every Boolean function has weak PTF density at most O(1)2ⁿ. This gives a negative answer to a question posed by Saks [1993]. 4) PTF degree └log₂m┘+1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [2000].