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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 Abbe, Shpilka, and Wigderson on Reed--Muller codes.<\/jats:p>","DOI":"10.1137\/19m1257792","type":"journal-article","created":{"date-parts":[[2019,10,8]],"date-time":"2019-10-08T10:35:49Z","timestamp":1570530949000},"page":"699-729","source":"Crossref","is-referenced-by-count":14,"title":["Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors"],"prefix":"10.1137","volume":"1","author":[{"given":"Pierre","family":"Baldi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Roman","family":"Vershynin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,10,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","first-page":"5229","DOI":"10.1109\/TIT.2015.2462817","volume":"61","author":"Abbe E.","year":"2015","journal-title":"IEEE Trans. Inform. 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