Minimal integral of squared Lagrange fundamental polynomials: is $\min I = 2-(1+o(1))/n$? (Erdős #1131)
Statement
For distinct nodes $x_1,\ldots,x_n\in [-1,1]$ let $$l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)}$$ be the Lagrange fundamental polynomials, which satisfy $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$. What is the minimal value, over all choices of nodes, of $$I(x_1,\ldots,x_n)=\int_{-1}^1 \sum_k \lvert l_k(x)\rvert^2\,\mathrm{d}x?$$ In particular, is it true that $$\min I = 2-(1+o(1))\frac{1}{n}?$$
Acceptance. FULLY RESOLVES: a proof determining the asymptotics of $\min I$ — establishing $\min I = 2-(1+o(1))/n$, or refuting it by proving $\min I = 2 - \omega(1/n)$ error (e.g. a matching lower-bound construction showing the error term is of order $(\log n)^a/n$ for some $a>0$). Machine-checkable (Lean/Coq) preferred, else a complete written proof. ADVANCES: (a) an improved lower bound — strictly better than the $2-O((\log n)^2/n)$ bound stated in the background (e.g. $2-O(\log n/n)$ or $2-O(1/n)$) — with proof; (b) an improved upper bound strictly below the $2-2/(2n-1)$ bound stated in the background, with an explicit node construction and proof; (c) rigorously certified values (interval-arithmetic enclosures with certified global optimality or certified local-minimizer analysis) of $\min I$ for a range of $n$ wide enough to empirically discriminate the two candidate scalings, with reproducible code. Deliver the proof file, or the optimization/certification code plus the table of certified enclosures.
Background
Posed by Erdős [Er61, p.67], revisited in [Er95e], and recorded in the problem collection [Va99, 2.45]; listed as open on erdosproblems.com/1131 (fetched 2026-07-13, status 'open', tagged 'analysis | polynomials'). History: Fejér [Fe32] proved that the roots of the integral of the Legendre polynomial minimize the sup-norm analogue $\max_{x\in[-1,1]}\sum_k \lvert l_k(x)\rvert^2$, and Erdős initially conjectured the same nodes minimize the integral $I$; Szabados [Sz66] disproved that for every $n>3$. The known frontier is due to Erdős, Szabados, Varma, and Vértesi [ESVV94]: $$2-O\left(\frac{(\log n)^2}{n}\right)\leq \min I\leq 2-\frac{2}{2n-1},$$ where the upper bound is witnessed by the Fejér nodes above. The open gap is thus between error terms of order $(\log n)^2/n$ and $1/n$; the conjecture asserts the truth is $2-(1+o(1))/n$. Caution: as of the fetch, the problem page's comment banner records claims of partial results in the (login-gated) forum thread, and the teorth/erdosproblems AI-contributions registry lists a candidate partial result obtained with GPT-5.5 Pro (April 2026); neither has been incorporated into the site's remarks, so verify the current frontier before investing. The attacker's tool: high-precision numerical optimization over node configurations — $I$ is an explicit rational-in-the-nodes objective computable exactly by Gauss–Legendre quadrature, so certified enclosures of $\min I$ for concrete $n$ can discriminate the $(\log n)^2/n$ vs $1/n$ scaling — combined with hard-analysis sharpening of the [ESVV94] lower bound.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1131 (T. F. Bloom) | website |
| REF-02 | AI contributions to Erdős problems (teorth/erdosproblems wiki) — records a candidate partial result on #1131 | website |
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