Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors
Skip to main content
Open Access Publications from the University of California

UC Irvine

UC Irvine Previously Published Works bannerUC Irvine

Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors


A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The partial case of this problem for degree $d=1$ was solved by Zuev in 1989, who showed that the number $T(n,1)$ of linear threshold functions satisfies $\log_2 T(n,1) \approx n^2$, up to smaller order terms. However the number of polynomial threshold functions for any higher degrees, including $d=2$, has remained open. We settle this problem for all fixed degrees $d \ge1$, showing that $ \log_2 T(n,d) \approx n \binom{n}{\le d}$. The solution relies on connections between the theory of Boolean threshold functions, hyperplane arrangements, and random tensors. Perhaps surprisingly, it uses also a recent result of E.Abbe, A.Shpilka, and A.Wigderson on Reed-Muller codes.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View