FormalConjectures.ErdosProblems.«1150»

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import FormalConjecturesUtil

Erdős Problem 1150

Reference: erdosproblems.com/1150


open scoped Polynomial


namespace Erdos1150

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

@[category research open, AMS 12 30]
theorem declaration uses 'sorry'erdos_1150 :
    answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in Filter.atTop,
      ∀ P : ℂ[X],  (∀ i ≤ P.natDegree, P.coeff i = - 1 ∨ P.coeff i = 1) → P.natDegree = n →
        ⨆ z : Metric.sphere (0 : ℂ) 1, ‖P.eval (z : ℂ)‖ > (1 + c) * Real.sqrt n := by⊢ True ↔
  ∃ c > 0,
    ∀ᶠ (n : ℕ) in Filter.atTop,
      ∀ (P : ℂ[X]),
        (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) →
          P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ > (1 + c) * √↑n
  sorryAll goals completed! 🐙

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 $|a_k|^2 = 1$.

@[category textbook, AMS 12 30]
theorem declaration uses 'sorry'erdos_1150.variants.parseval_lower_bound (P : ℂ[X]) (n : ℕ)
    (hcoeff : ∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1)
    (hdeg : P.natDegree = n) :
    ⨆ z : Metric.sphere (0 : ℂ) 1, ‖P.eval (z : ℂ)‖ ≥ Real.sqrt (n + 1) := byP:ℂ[X]n:ℕhcoeff:∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1hdeg:P.natDegree = n⊢ ⨆ z, ‖Polynomial.eval (↑z) P‖ ≥ √(↑n + 1)
  sorryAll goals completed! 🐙


end Erdos1150