DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Regular holomorphic germs in the last variable

Definition

Fix $m \geq 1$ , write $z = (z^{'}, z_{m})$ with $z^{'} = (z_{1}, \dots, z_{m - 1}) \in C^{m - 1}$ , and let $f \in O_{m, 0}$ . For $d \in N$ , the germ $f$ is regular in $z_{m}$ of order $d$ if some representative satisfies

$f (0, ζ) = ζ^{d} h (ζ)$

on a neighbourhood of $0 \in C$ , where $h$ is holomorphic and $h (0) \neq 0$ . Equivalently, the one-variable slice $ζ \mapsto f (0, ζ)$ has a zero of exact order $d$ at the origin.

The case $d = 0$ is exactly the unit case $f (0) \neq 0$ .

Depends on

The ring of holomorphic germs at $0$ and its maximal ideal

Regular holomorphic germs in the last variable

Definition

Depends on

Dependency tree · 2 levels

Sources