DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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.

An affine algebraic set in affine space

Definition

Let $k$ be an algebraically closed field and let $S \subseteq k [x_{1}, \dots, x_{n}]$ . The affine algebraic set cut out by $S$ is $V (S) : = {a = (a_{1}, \dots, a_{n}) \in k^{n} : f (a) = 0 for every f \in S} .$ We also write $A_{k}^{n} : = k^{n}$ .

By convention, $V (\emptyset) = A_{k}^{n} and V (1) = \emptyset,$ because every point vacuously annihilates every polynomial in the empty set, while no point annihilates the constant polynomial $1$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

J. S. Milne, Algebraic Geometry, Chapter 2a (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, §1.1 (standard reference, not scraped)