CounterexampleConstruction: AI-generatedVerification: 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.

inhomogeneous equation not projectively well defined

Statement refuted

An arbitrary polynomial equation has a representative-independent zero locus in projective space.

Counterexample

Given: A field $k$ of characteristic not equal to $2$ , the inhomogeneous polynomial $x_{0} + 1$ , and the projective point $[1 : 0] \in P_{k}^{1}$ .

1.1givenalgebra

The vectors $(1, 0)$ and $(- 1, 0)$ represent the same projective point, but $x_{0} + 1$ takes the nonzero value $2$ at the first and the value $0$ at the second.

2.1step 1.1∎

Thus the vanishing decision changes with the representative, refuting the assertion.

Depends on

Used by

Nothing in the library uses this result yet.

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)