LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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A finite coefficient vector has a canonical polynomial representative

Statement

For every $n \in N$ and reals $c_{0}, \dots, c_{n}$ , the function $q (x) = \sum_{k = 0}^{n} c_{k} x^{k}$ is the evaluation of a formal real polynomial of degree at most $n$ . The representative can be chosen so that it is the zero polynomial exactly when every $c_{k} = 0$ ; if $c_{n} \neq 0$ , it has degree $n$ and leading coefficient $c_{n}$ . Here the coefficient vector may be padded by zeros for the displayed sum; padding describes the evaluation, not a change to the formal polynomial object.

Facts & Assumptions

Given: A natural $n$ and a finite real coefficient vector $c_{0}, \dots, c_{n}$ .

[F1]

Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials: a nonzero formal polynomial is a finite coefficient list whose final coefficient is nonzero; its evaluation is the corresponding finite sum, and the zero polynomial is a separate object.

[F2]

Finite sums and finite products, by recursion and Laws of finite sums and finite products: the empty sum is zero, a finite sum extends by its last term, and the finite-sum notation agrees with that recursion.

[F3]

The principle of mathematical induction: a property true at zero and preserved by successors holds for every natural number.

[F4]

The reals form a field and Multiplication by zero: $0 \cdot a = 0$ : real addition and multiplication have the field identities, including $0 x = 0$ .

Proof

technique · induction

1.1basegivenF1F4algebraconstruct

For $n = 0$ , if $c_{0} = 0$ choose the zero polynomial, whose evaluation is $0$ ; if $c_{0} \neq 0$ , choose the one-term formal list $(c_{0})$ , of degree $0$ and leading coefficient $c_{0}$ . In both cases its evaluation is $c_{0} x^{0} = c_{0}$ .

1.2givenF1F2F4algebraihconstruct

Suppose the assertion holds for vectors of length $n + 1$ , and consider $c_{0}, \dots, c_{n}, c_{n + 1}$ . If $c_{n + 1} = 0$ , the finite-sum recursion and [F4] show that the displayed evaluation is the evaluation for the prefix vector; use its representative from the induction hypothesis. If $c_{n + 1} \neq 0$ , the formal coefficient list $(c_{0}, \dots, c_{n}, c_{n + 1})$ is valid by [F1], has degree $n + 1$ and leading coefficient $c_{n + 1}$ , and evaluates to the displayed sum by [F2]. The zero-characterization in the first case is inherited from the prefix; in the second case both the vector and the polynomial are nonzero.

2.1step 1.1step 1.2F3discharge-induction∎

Induction [F3] now supplies the representative for every $n$ . The construction also proves that it is zero exactly when all vector entries vanish, and that a nonzero final entry is its leading coefficient at degree $n$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Mark Embree, Lecture Notes on Numerical Analysis, lecture 1 (standard reference, not scraped)
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation (standard reference, not scraped)