LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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 polynomial signs force the Haar zero-count contradiction

Statement

Let $n \geq 0$ , let $p$ be a nonzero real polynomial of degree at most $n$ , and let $x_{0} < \dots < x_{n + 1}$ be real numbers. The values $p (x_{i})$ cannot all be nonzero with alternating signs.

Facts & Assumptions

Given: $n \geq 0$ , a nonzero real polynomial $p$ of degree at most $n$ , and $x_{0} < \dots < x_{n + 1}$ with each $p (x_{i}) \neq 0$ .

[F1]

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: every real polynomial function is continuous on its domain.

[F2]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : if a continuous real function has endpoint values of opposite signs on a closed interval, it has a zero in the open interval.

[F3]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a nonzero real polynomial of degree at most $n$ has at most $n$ distinct real roots.

Proof

1.1givenF1F2construct

Suppose that the nonzero values $p (x_{i})$ alternate in sign. By [F1], $p$ is continuous on each $[x_{i}, x_{i + 1}]$ . Since the endpoint values are nonzero and have opposite signs, [F2] gives a root $z_{i} \in (x_{i}, x_{i + 1})$ for every $0 \leq i \leq n$ . These open intervals are pairwise disjoint, so the $z_{i}$ are $n + 1$ distinct real roots.

2.1givenF3step 1.1algebra∎

The $n + 1$ distinct roots from step 1.1 contradict [F3], because $p$ is nonzero and has degree at most $n$ . Therefore the assumed alternating nonzero values cannot occur.

Depends on

Used by

Dependency tree · two levels

30 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

Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapter 10 (standard reference, not scraped)
András Kroó and Allan Pinkus, Strong Uniqueness, §4.4 (standard reference, not scraped)