False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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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.
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A holomorphic function on a domain in $C^{2}$ vanishing on a set with an accumulation point vanishes identically

Statement

False claim: if $f$ is holomorphic on a nonempty connected open subset of $C^{2}$ and the zero set of $f$ has an accumulation point in the domain, then $f$ is identically zero.

This is exactly the one-variable identity theorem carried over without change. The several-variable page proves only the honest open-set form (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

Facts & Assumptions

Given: The false claim above and the function $f (z_{0}, z_{1}) = z_{0}$ on $C^{2}$ .

[L1]

In one complex variable, equality on a set with an accumulation point forces equality everywhere (Identity theorem for holomorphic functions).

[L2]

In one complex variable, a nonzero holomorphic function has isolated zeros (Zeros of a nonzero holomorphic function are isolated).

[L3]

The function $f (z_{0}, z_{1}) = z_{0}$ is a nonzero holomorphic function on $C^{2}$ whose zero set is the hyperplane ${0} \times C$ , hence is unbounded and has no isolated points (A nonzero holomorphic function on $C^{2}$ whose zero set is an unbounded hyperplane).

Refutation

technique · direct

1.1

The false claim is the one-variable identity theorem [L1] repeated verbatim in two complex variables.

1.2

By [L3], the function $f (z_{0}, z_{1}) = z_{0}$ is holomorphic and not identically zero, while every point of its zero set ${0} \times C$ is an accumulation point of that zero set. So the hypothesis of the false claim holds for this $f$ .

2.1

Yet the conclusion fails, since $f (1, 0) = 1 \neq 0$ . What breaks from the one-variable proof is exactly [L2]: in one variable a nonzero holomorphic function has isolated zeros, whereas [L3] shows that in several variables a nonzero holomorphic function can vanish on a whole hyperplane. Therefore the false claim is false.

step 1.2L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

J. Lebl, Tasty Bits of Several Complex Variables, v4.4, §1.2 (standard reference, not scraped)
H. P. Boas, Lecture Notes on Multidimensional Complex Analysis, Ch. 2 (standard reference, not scraped)

A holomorphic function on a domain in C2C2 vanishing on a set with an accumulation point vanishes identically