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declared falsifiable · $250erdos:114
Erdős problem 114
polynomialsanalysisN/A
declared proved (Lean) · formalizederdos:115
True ↔ ∀ ε > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (p : Polynomial ℂ), p.Monic → p.natDegree = n → IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} → ∀ (z : ℂ), ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2
polynomialsanalysisN/A5 sources
declared provederdos:116
Erdős problem 116
polynomialsanalysisN/A
declared solved · formalized · $100erdos:119
For unit-modulus complex numbers ziz_i, let pn(z)=∏i≤n(z−zi)p_n(z)=\prod_{i\le n}(z-z_i) and Mn=max⁡∣z∣=1∣pn(z)∣M_n=\max_{|z|=1}|p_n(z)|. Erdős's prize question: is there c>0c>0 with ∑k≤nMk>n1+c\sum_{k\le n} M_k > n^{1+c}?
analysispolynomialsN/A4 sources
declared proved · formalizederdos:228
True ↔ ∃ c₁ c₂, ∀ᶠ (n : ℕ) in Filter.atTop, ∃ p, p.degree = ↑n ∧ (∀ i ≤ n, p.coeff i = 1 ∨ p.coeff i = -1) ∧ ∀ (z : ℂ), ‖z‖ = 1 → √↑n < c₁ * ‖Polynomial.eval z p‖ ∧ ‖Polynomial.eval z p‖ < c₂ * √↑n
analysispolynomialsN/A2 sources
declared disprovederdos:230
Erdős problem 230
analysispolynomialsN/A
declared provederdos:485
Erdős problem 485
analysispolynomialspossible2 sources
declared open · formalizederdos:521
False ↔ Erdos521.Claim
analysispolynomialsprobabilityN/A4 sources
declared open · formalizederdos:522
For Pn(z)=∑k=0nεkzkP_n(z) = \sum_{k=0}^n \varepsilon_k z^k with independent uniform signs, does the number RnR_n of roots in ∣z∣≤1|z| \le 1 satisfy Rn/(n/2)→1R_n/(n/2) \to 1 almost surely? The manuscript proves the strong law with Rn=n/2+Oω(n149/150)R_n = n/2 + O_\omega(n^{149/150}).
analysispolynomialsprobabilityN/A4 sources
declared provederdos:523
Erdős problem 523
analysisprobabilitypolynomialsN/A
declared openerdos:524
Erdős problem 524
analysisprobabilitypolynomialsN/A3 sources
declared provederdos:525
Erdős problem 525
analysisprobabilitypolynomialspossible
declared provederdos:1114
Erdős problem 1114
analysispolynomialsN/A
declared provederdos:1129
Erdős problem 1129
analysispolynomialsN/A2 sources
declared provederdos:1130
Erdős problem 1130
analysispolynomialsN/A2 sources
declared openerdos:1131
Erdős problem 1131
analysispolynomialsN/A2 sources
declared openerdos:1132
Erdős problem 1132
analysispolynomialsN/A2 sources
declared open · formalizederdos:1133
Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below (1+ε)n(1+\varepsilon)n to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.
analysispolynomialsN/A4 sources
declared open · formalizederdos:1150
True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (P : Polynomial ℂ), (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) → P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ > (1 + c) * √↑n
analysispolynomialsN/A2 sources
declared openerdos:1151
Let Lnf\mathcal{L}^nf be the Lagrange interpolation polynomials of a continuous ff on the Chebyshev nodes. Prove that, for any closed A⊆[−1,1]A\subseteq [-1,1], there exists a continuous function ff such that AA is the set of limit points of Lnf(x)\mathcal{L}^nf(x).
analysispolynomialsN/A3 sources
declared openerdos:1152
Erdős problem 1152
analysispolynomialsN/A
declared provederdos:1153
Erdős problem 1153
analysispolynomialsN/A3 sources

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