Erdős problem 1150 · problems.science

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$ , $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

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FormalConjectures/ErdosProblems/

1150.lean

Retained formal statement2 of 2

The trivial lower bound from Parseval's identity: for any polynomial $P$ of degree $n$ with coefficients in $\{-1, 1\}$ , we have $\max_{|z|=1} |P(z)| \geq \sqrt{n+1}$ .

This follows from Parseval's identity: $\frac{1}{2\pi} \int_0^{2\pi} |P(e^{i\theta})|^2 d\theta = \sum_{k=0}^{n} |a_k|^2 = n+1$ since each $|a_k|^2 = 1$ .

FormalConjectures/ErdosProblems/1150.leanErdos1150.erdos_1150.variants.parseval_lower_bound2 linesExact file

1∀ (P : Polynomial ℂ) (n : ℕ),2  (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) → P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ ≥ √(↑n + 1)

TextbookStatement only, no proof

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