DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-11

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.

Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree

Definition

Let $0 \neq f = \sum_{i} a_{i} x^{i} \in R [x]$ . Its degree is the largest natural number $n$ for which $a_{n} \neq 0$ , and its leading coefficient is $a_{n}$ :

$\deg f : = \max {i \in N : a_{i} \neq 0}, lc (f) : = a_{\deg f} .$

The maximum exists because the nonempty support is finite. The polynomial $f$ is monic when $lc (f) = 1_{R}$ . The zero polynomial has no degree and no leading coefficient. Accordingly, every degree statement below explicitly separates the zero polynomial rather than assigning it a formal degree.

Depends on

The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree

Definition

Depends on

Dependency tree · 2 levels

Sources