LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

homogenization dehomogenization correspondence

Statement

If $f \in k [x_{1}, \dots, x_{n}]$ has degree at most $d$ , then $f^{h} = x_{0}^{d} f (x_{1} / x_{0}, \dots, x_{n} / x_{0})$ is homogeneous of degree $d$ and $(f^{h}) (1, x) = f (x)$ . If $F$ is homogeneous of degree $d$ , then $F = x_{0}^{d} F (1, x_{1} / x_{0}, \dots, x_{n} / x_{0})$ .

Proof

Given: $f = \sum_{∣ α ∣ \leq d} c_{α} x^{α}$ and homogeneous $F$ of degree $d$ .

1.1

$f^{h} = \sum c_{α} x_{0}^{d - ∣ α ∣} x^{α}$ , so each monomial has degree $d$ and $x_{0} = 1$ recovers $f$ .

givenalgebra

2.1

Factoring $x_{0}^{d}$ from every homogeneous monomial of $F$ proves the displayed reverse formula.

givenalgebra∎

Depends on

homogeneous polynomial and homogeneous ideal

Used by

projective closure affine set Definition
projective hypersurface affine pieces Lemma
global regular functions projective variety Theorem
ideal projective closure saturation Theorem

Dependency tree · two levels

3 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, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)