DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

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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.

Depends on

Used by

Dependency tree · two levels

4 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, Sections 4.3-4.4 (standard reference, not scraped)