False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

FALSE: arbitrary factorizations by a Weierstrass polynomial are unique without unit and degree conditions

Statement

False claim: For a regular germ $f$ , a factorization $f = a P$ is unique whenever $P$ is a Weierstrass polynomial, even if $a$ need not be a unit and the degree of $P$ need not equal the regular order of $f$ .

Facts & Assumptions

Given: The germ $f (z_{1}, z_{2}) = z_{2}^{2}$ .

[L1]

The genuine Weierstrass preparation theorem factors a regular germ as a unit times a Weierstrass polynomial (Weierstrass preparation theorem).

[L2]

Both $z_{2}$ and $z_{2}^{2}$ are Weierstrass polynomials in the variable $z_{2}$ (Weierstrass polynomials in the last variable).

Refutation

technique · direct

1.1

The factorization $z_{2}^{2} = 1 \cdot z_{2}^{2}$ is a genuine preparation by [L1].

L1given

2.1

If neither the unit condition nor the degree condition is retained, then also $z_{2}^{2} = z_{2} \cdot z_{2}$ is allowed, and [L2] says the second factor is a Weierstrass polynomial of degree $1$ . This differs from the genuine degree- $2$ preparation in step 1.1, so the relaxed factorization is not unique.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Jiří Lebl, Tasty Bits of Several Complex Variables, Section 6.2 (standard reference, not scraped)