Problem 1130

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claims/: The 1 claim page of Problem 1130, one per claimant's result; the problem's standing derives from them.

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Statement. For $x_1,\ldots,x_n\in [-1,1]$ let

$$l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$$

which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.

Let $x_0=-1$ and $x_{n+1}=1$ and

$$\Upsilon(x_1,\ldots,x_n)=\min_{0\leq i\leq n}\max_{x\in[x_i,x_{i+1}]} \sum_k \lvert l_k(x)\rvert.$$

Is it true that

$$\Upsilon(x_1,\ldots,x_n)\ll \log n?$$

Describe which choice of $x_i$ maximise $\Upsilon(x_1,\ldots,x_n)$.

Status. The site labels the problem PROVED (page last edited 17 January 2026, label accessed 2026-09-04). The site records that de Boor and Pinkus [dBPi78] proved Erdős's conjectured characterization of the maximizing nodes, from which the logarithmic bound follows; the accepted claim page is de Boor and Pinkus 1978, which records the paper's convention that the endpoints are nodes. The problem pairs a yes-or-no question, answered yes, with a request to describe the maximizing choice, so the derived claim value is answered.

Source. erdosproblems.com/1130, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1130, https://www.erdosproblems.com/1130.

References.

• [Er47] Erdős, P., Some remarks on polynomials. Bull. Amer. Math. Soc. 53 (1947), 1169-1176.
• [dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of Bernstein and Erdős concerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289-303.

Formalization. No formal-conjectures statement file exists for the problem, and the site's page reports no formalized statement. A third-party Lean 4 file in the lean-proofs repository, which names de Boor and Pinkus as its informal authors and formalizes the literal free-node formulation, proves for three nodes that the equal-maxima characterization fails for free nodes; the claim page for de Boor and Pinkus 1978 links it at its pinned commit. It is not built or audited in this repository.

Current assessment

The question (site formulation, page last edited 17 January 2026). For nodes $x_1,\ldots,x_n$ ranging over $[-1,1]$, with $x_0=-1$ and $x_{n+1}=1$, let $\Upsilon$ be the least of the $n+1$ maxima of the Lebesgue function $\sum_k|l_k(x)|$ over the pieces $[x_i,x_{i+1}]$. Is $\Upsilon\ll\log n$, and which choice of nodes maximizes $\Upsilon$? PROVED. The site's commentary records Erdős's bound $\Upsilon<\sqrt n$ from [Er47], his expectation that the maximum is attained when all $n+1$ piece maxima are equal, the same characterization as on Problem 1129, and that de Boor and Pinkus proved it, so that $\Upsilon\le(2/\pi)\log n+O(1)$ follows from the bounds discussed on Problem 1129. The site's wording, with free nodes and $n+1$ pieces, is how Erdős posed the question [Er47, pp. 1171-1172], and the standing answers it. De Boor and Pinkus work with systems containing both endpoints, whose pieces are the $n-1$ gaps between the $n$ nodes. For those systems their Theorem 2 gives $\min_i\lambda_i\le\lambda^*\le\max_i\lambda_i$, with $\lambda^*$ the least Lebesgue constant of $n$ nodes, so the least gap maximum is largest for the unique equioscillating system $t^*$ alone. Both questions transfer to the site's wording through an affine rescaling, an elementary step recorded on the claim page and not independently reviewed. The interior gaps of any system are the gaps of its rescaling onto $[-1,1]$. So $\Upsilon\le\lambda^*\le(2/\pi)\log n+O(1)$, with equality exactly for the affine images of $t^*$ inside $[-1,1]$ whose two end-piece maxima are at least $\lambda^*$. The symmetric image whose end-piece maxima equal $\lambda^*$ has all $n+1$ piece maxima equal, so the maximum is attained there, as Erdős expected. But equal maxima do not characterize the maximizers: images whose end-piece maxima exceed $\lambda^*$ attain it too. For three nodes, $(-1/2,0,1/2)$ is a maximizer with piece maxima $7,5/4,5/4,7$. For $n\ge3$ the system $t^*$ itself, whose end pieces are points with maximum $1$, is not a maximizer.

Standing. One accepted full claim, de Boor and Pinkus 1978, refereed in Journal of Approximation Theory 24 (1978), no. 4, 289–303, and credited by the site's curator. The first question is answered yes, with $\Upsilon\le\lambda^*\le(2/\pi)\log n+O(1)$ through the Chebyshev nodes; the second is answered in the site's convention through the rescaling recorded on the claim page; the derived claim value is answered. The theorem statements are checked against the paper, not its proofs in detail; the proofs are not compiled in this wiki.

Formalization. No formal-conjectures statement file exists for the problem. The file Erdos1130.lean in the lean-proofs repository, with de Boor and Pinkus as informal authors and Codex and GPT-5.6 Sol as formal authors, declares itself a formalization of the literal free-node formulation and proves for three nodes that $\Upsilon\le5/4$ and that the maximizer $(-1/2,0,1/2)$ has piece maxima $7,5/4,5/4,7$, not all equal; the claim page links it at its pinned commit. It is not built or audited in this repository, so no formalized evidence is listed.

Search scope. The site's problem page and its proof-claims tab, which carries no claim; the community database at teorth/erdosproblems, which lists the problem as proved and unformalized; the formal-conjectures tree; the lean-proofs file at its pinned commit; de Boor and Pinkus 1978.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• deboor_1978_conjectures_bernstein_erdos_optimal_nodes_polynomial_interpolation
• erdos_1947_remarks_polynomials
• erdos_1947_remarks_polynomials / theorem_2
• erdos_1967_problems_results_convergence_divergence_properties_lagrange
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / problem_p66
• vertesi_2013_paul_erdos_interpolation_problems_results_new
• vertesi_2013_paul_erdos_interpolation_problems_results_new / theorem_2_4