Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 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≠f=∑iaixi∈R[x]0\ne f=\sum_i a_i x^i\in R[x]. Its degree is the largest natural number nn for which an≠0a_n\ne0, and its leading coefficient is ana_n:

deg⁡f:=max⁡{i∈N:ai≠0},lc⁡(f):=adeg⁡f.\deg f:=\max\{i\in\mathbb N:a_i\ne0\},\qquad \operatorname{lc}(f):=a_{\deg f}.

The maximum exists because the nonempty support is finite. The polynomial ff is monic when lc⁡(f)=1R\operatorname{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

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources