Problem 1130

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 $\Upsilon(x_1,\ldots,x_n)$ .

Source: erdosproblems.com/1130

Analysis
Polynomials

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

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.

Problem 1130

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