DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

The classical vanishing ideal

Definition

For any subset $E \subseteq k^{n}$ , define $I (E) = {f \in k [x_{1}, \dots, x_{n}] : f (a) = 0 for every a \in E} .$ This is the vanishing ideal. Indeed $0 \in I (E)$ , $(f - g) (a) = 0 - 0 = 0$ for $f, g \in I (E)$ , and $(r f) (a) = r (a) 0 = 0$ for any polynomial $r$ . These verify Ideal criteria and intersections of ideals. Also $f^{m} \in I (E)$ , $m \geq 1$ , implies $f (a)^{m} = 0$ and hence $f (a) = 0$ for every $a \in E$ ; thus $I (E)$ is radical. Vacuous quantification gives $I (\emptyset) = R$ .

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §2e pp. 40–42. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Dependency tree · two levels

8 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

J. S. Milne, Algebraic Geometry v6.10, §2e pp. 40–42 (standard reference, not scraped)