OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d\sqrt n, uniformly for 1≤ d≤ n. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with \mathop{sign}\nolimits (0)=1.
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For every n ≥ 1 and 1≤ d≤ n, we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most 8d\sqrt n. This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention \mathop{sgn}\nolimits (0)=1.
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