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

Used by

z₁z₂ is not regular in z₂ at the origin Counterexample
Weierstrass polynomials in the last variable Definition
A linear shear makes z₁z₂ regular in z₂ Example
After a linear coordinate change, every nonzero germ is regular in the last variable Lemma
Nearby slices of a regular germ have the same zero count Lemma
Weierstrass preparation theorem 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

Jiří Lebl, Tasty Bits of Several Complex Variables, Section 6.2 (standard reference, not scraped)
Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Section 4.1 (standard reference, not scraped)