TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically

Statement

Let $m \geq 1$ , let $U \subseteq C^{m}$ be a nonempty connected open set, and let $f : U \to C$ be holomorphic (Holomorphic functions on an open subset of $C^{m}$ ). If there is a nonempty open set $W \subseteq U$ such that $f (z) = 0$ for every $z \in W$ , then $f \equiv 0$ on $U$ .

This is the several-variable identity theorem at the strength the page supports: the hypothesis is a nonempty open set of zeros. An accumulation point of the zero set is neither assumed nor sufficient in several variables; the companion page records that stronger one-variable statement as false here.

Facts & Assumptions

Given: A nonempty connected open set $U \subseteq C^{m}$ , a holomorphic function $f : U \to C$ , and a nonempty open set $W \subseteq U$ on which $f = 0$ .

[L1]

Holomorphic functions of several variables are smooth; every point $a \in U$ has a polydisc $Δ_{r} (a) \subseteq U$ on which $f (z) = \sum_{α} c_{α} (z - a)^{α}, c_{α} = \frac{\partial_{z}^{α} f (a)}{α!};$ and all mixed complex derivatives are holomorphic (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic).

[L2]

A connected space has no nontrivial clopen subsets (For a topological space the following agree: no separation exists, the only clopen subsets are $\emptyset$ and $X$ , and every continuous map to the two-point discrete space is constant, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[L3]

The mixed complex derivative notation $\partial_{z}^{α}$ and the zero-order identity $\partial_{z}^{0} f = f$ are those in the power-series and smoothness statement [L1]; $C^{k}$ maps and multi-index derivative notation in Euclidean space supplies the underlying multi-index arithmetic.

[L4]

Polydiscs in $C^{m}$ are the coordinatewise discs of Balls, polydiscs and the distinguished boundary in $C^{m}$ .

Proof

technique · direct

1.1givenL1L3

For every multi-index $α$ , the derivative $\partial_{z}^{α} f$ is holomorphic and therefore continuous on $U$ by [L1]; since $f = 0$ on the open set $W$ , every derivative of $f$ is also $0$ on $W$ , so the set $A : = {z \in U : \partial_{z}^{α} f (z) = 0 for every α}$ contains $W$ and is therefore nonempty.

2.1step 1.1

The set $A$ is closed in $U$ , because it is the intersection over all multi-indices $α$ of the closed zero sets of the continuous functions $\partial_{z}^{α} f$ .

2.2step 1.1L1L4

The set $A$ is open in $U$ : if $a \in A$ , choose a smaller polydisc $Δ \subseteq U$ centred at $a$ ; then every coefficient $\partial^{α} f (a) / α!$ in the power-series expansion of $f$ on $Δ$ is $0$ , so [L1] gives $f = 0$ on $Δ$ , and hence every derivative vanishes on $Δ$ as well, which means $Δ \subseteq A$ .

3.1step 2.1step 2.2L2L3∎

The set $A$ is a nonempty subset of $U$ that is both open in $U$ and closed in $U$ , so connectedness and [L2] force $A = U$ ; in particular $f = \partial_{z}^{0} f$ vanishes at every point of $U$ , hence $f \equiv 0$ on $U$ .

Remarks

Why the hypothesis is open-set vanishing and not an accumulation point. In one complex variable, accumulation of zeros implies equality by local factorisation and isolated zeros. In several variables the zero set of a nonzero holomorphic function can contain whole complex hypersurfaces, so the open-set hypothesis is the honest form at this stage.
What the proof really uses. The proof needs only two page-level tools: holomorphic smoothness and the local power-series expansion. Once every derivative at one point vanishes, the power series on a smaller polydisc is identically zero, and connectedness propagates that local vanishing to the whole set.

Depends on

Used by

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Sources

J. Lebl, Tasty Bits of Several Complex Variables, v4.4, Thm. 1.2.7 (standard reference, not scraped)
H. P. Boas, Lecture Notes on Multidimensional Complex Analysis, Ch. 2 (standard reference, not scraped)
M. Jabbari, Notes for Analysis and Geometry of Several Complex Variables, Thm. 22(7) (standard reference, not scraped)