OPEN
This is open, and cannot be resolved with a finite computation.
Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$,\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?\]
In other words, does there exist an 'ultraflat' polynomial with coefficients $\pm 1$. The answer is yes if the coefficients can take any values on the unit circle (see
[230]).
The lower bound\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert\geq \sqrt{n}\]is trivial from Parseval's theorem.
A weaker 'flatness' question is the subject of
[228].
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This page was last edited 23 January 2026.
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