LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Alternating interpolation weights annihilate lower-degree polynomials

Statement

Let $N \geq 0$ and let $x_{0} < \dots < x_{N + 1}$ be real nodes. Set $ω (t) = \prod_{j = 0}^{N + 1} (t - x_{j})$ and $w_{i} = \prod_{j \neq i} (x_{i} - x_{j})$ . If $g$ is a real polynomial of degree at most $N$ , then $\sum_{i = 0}^{N + 1} \frac{g (x_{i})}{w_{i}} = 0.$ If, for one fixed $ε \in {- 1, 1}$ , the values $ε (- 1)^{i} g (x_{i})$ are all nonnegative, then $g$ is the zero polynomial.

Facts & Assumptions

Given: $N \geq 0$ , strictly increasing real nodes $x_{0}, \dots, x_{N + 1}$ , and $g \in P_{N} (R)$ .

[F1]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a nonzero polynomial of degree at most $N + 1$ cannot vanish at $N + 2$ distinct nodes; a nonzero polynomial of degree at most $N$ cannot vanish at $N + 1$ distinct nodes.

Proof

1.1givenF1constructalgebra

Each $w_{i} \neq 0$ . The Lagrange polynomial $L_{i} (t) : = ω (t) / ((t - x_{i}) w_{i})$ has degree $N + 1$ and values $L_{i} (x_{j}) = 1$ for $j = i$ and $0$ otherwise. Thus $H (t) : = \sum_{i} g (x_{i}) L_{i} (t)$ has degree at most $N + 1$ and agrees with $g$ at all $N + 2$ nodes. By [F1], $H - g = 0$ , so $H = g$ .

2.1givenstep 1.1algebra

The coefficient of $t^{N + 1}$ in $L_{i}$ is $1 / w_{i}$ , while that coefficient in $g$ is $0$ . Comparing the $t^{N + 1}$ coefficients in $H = g$ from step 1.1 proves the displayed sum identity.

3.1givenstep 2.1algebra

The sign of $w_{i}$ is $(- 1)^{N + 1 - i}$ , since exactly the $N + 1 - i$ factors with $j > i$ are negative. Hence all terms $g (x_{i}) / w_{i}$ have one common weak sign when $ε (- 1)^{i} g (x_{i}) \geq 0$ . Their sum is $0$ by step 2.1, so every term is $0$ , and all $N + 2$ values $g (x_{i})$ vanish.

4.1givenF1step 3.1algebra∎

Since $g$ has degree at most $N$ , its $N + 2$ distinct roots in step 3.1 force $g = 0$ by [F1].

Depends on

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots

Used by

Weighted rational minimizers satisfy the defect-sensitive alternation criterion Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

András Kroó and Allan Pinkus, Strong Uniqueness, §6.5 (standard reference, not scraped)
Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapter 24 (standard reference, not scraped)