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arXiv · 2610.34378

Logarithmic asymptotics for polynomial threshold functions

Martin Anthony
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Abstract

We count Boolean functions represented as the sign of a real polynomial of bounded degree on the $n$-dimensional cube. We prove that the logarithm of their number is asymptotic to that of the classical hyperplane-arrangement upper bound, uniformly over all degrees, with relative error $O((\log n)^2/n)$. We also obtain sharper additive and relative error estimates in particular degree ranges, including an additive error $O(n^2)$ between the logarithms throughout a small linear degree range and improved bounds just below the central degree. The proof combines probabilistic estimates for Reed--Muller codes with a deletion argument for sets of evaluation vectors, an entropy count, and a sharp-threshold estimate for expected rank deficiency.

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Martin Anthony. 2026-10-25. Logarithmic asymptotics for polynomial threshold functions. https://arxiv.org/abs/2610.34378

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