DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Weierstrass polynomials in the last variable

Definition

Fix $m \geq 1$ , write $z = (z^{'}, z_{m})$ with $z^{'} \in C^{m - 1}$ , and let $d \in N$ . A Weierstrass polynomial of degree $d$ is a germ in $O_{m, 0}$ represented by

$W (z^{'}, z_{m}) = z_{m}^{d} + a_{d - 1} (z^{'}) z_{m}^{d - 1} + \dots + a_{0} (z^{'}),$

where each $a_{j}$ is an element of $O_{m - 1, 0}$ , so for $m = 1$ the coefficients are complex constants, and $a_{j} (0) = 0$ .

For $d = 0$ the lower-coefficient list and sum are empty, so the unique degree- $0$ Weierstrass polynomial is $W = 1$ .

In particular $W (0, z_{m}) = z_{m}^{d}$ , so every Weierstrass polynomial of degree $d$ is regular in $z_{m}$ of order $d$ in the sense of Regular holomorphic germs in the last variable.

Weierstrass polynomials in the last variable

Definition

Depends on

Dependency tree · 2 levels

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