Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: separate holomorphy can fail to imply local boundedness

Statement

False claim: a separately holomorphic function on a finite-dimensional polydisc need not be locally bounded.

Facts & Assumptions

Given: A separately holomorphic function on a polydisc.

[L1]

Separate holomorphy forces boundedness on every smaller closed polydisc (Separate holomorphy forces local boundedness on smaller polydiscs).

Refutation

technique · direct
1.1

Let f be separately holomorphic on a polydisc.

given
2.1

The conclusion of [L1] applies directly to f, so f is locally bounded. This contradicts the displayed claim.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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