# Deterministic Approximate Counting for Degree-2 Polynomial Threshold Functions

We give a deterministic algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-2 polynomial threshold function. Given a degree-2 input polynomial p(x_1,...,x_n) and a parameter eps > 0, the algorithm approximates Pr[p(x)>=0] to within an additive \pm \eps in time poly(n,2^{\poly(1/\eps)}). (Note that it is NP-hard to determine whether the above probability is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms.) This is the first deterministic algorithm for this counting problem in which the running time is a fixed polynomial in n for any constant eps > 0. For "regular" polynomials p (those in which no individual variable's influence is large compared to the sum of all n variable influences) our algorithm runs in poly(n,1/eps) time.

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