Lower Bound on Weights of Large Degree Threshold Functions

An integer polynomial p of n variables is called a threshold gate for a Boolean function f of n variables if for all x ∈ {0, 1} f(x) = 1 if and only if p(x) > 0. The weight of a threshold gate is the sum of its absolute values. In this paper we study how large a weight might be needed if we fix some function and some threshold degree. We prove 2 ) lower bound on this value. The best previous bound was 2 ) (Podolskii, 2009). In addition we present substantially simpler proof of the weaker 2 ) lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.