ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 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.

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The holomorphic hull of a product torus in the bidisc is the closed polydisc it bounds

Example

Fix radii $0 < r_{1}, r_{2} < 1$ in the bidisc $D^{2}$ , and let

$T = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ = r_{1}, ∣ z_{2} ∣ = r_{2}} .$

Then the holomorphic hull of $T$ in $D^{2}$ is

${\hat{T}}_{D^{2}} = {(z_{1}, z_{2}) \in C^{2} : ∣ z_{1} ∣ \leq r_{1}, ∣ z_{2} ∣ \leq r_{2}} .$

Facts & Assumptions

Given: The product torus $T$ in the bidisc $D^{2}$ .

[L1]

Hull membership is tested against all holomorphic functions on the ambient domain (Holomorphic hulls and holomorphic convexity).

[L2]

On a closed polydisc, the supremum of a holomorphic function is attained on the distinguished boundary (The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary).

Verification

technique · direct

1.1L1L2given

Let $P = {(z_{1}, z_{2}) : ∣ z_{1} ∣ \leq r_{1}, ∣ z_{2} ∣ \leq r_{2}}$ . If $a \in P$ and $f$ is holomorphic on $D^{2}$ , then $f$ is continuous on the closed polydisc $P$ , and the distinguished boundary of $P$ is exactly $T$ . Therefore [L2] gives $∣ f (a) ∣ \leq \sup_{T} ∣ f ∣$ . By [L1], every point of $P$ lies in ${\hat{T}}_{D^{2}}$ .

2.1L1step 1.1algebra∎

If $a = (a_{1}, a_{2})$ lies in $D^{2} ∖ P$ , then either $∣ a_{1} ∣ > r_{1}$ or $∣ a_{2} ∣ > r_{2}$ . In the first case the holomorphic coordinate function $f (z_{1}, z_{2}) = z_{1}$ satisfies $∣ f (a) ∣ > \sup_{T} ∣ f ∣$ , and in the second case $f (z_{1}, z_{2}) = z_{2}$ does the same. Hence [L1] excludes every point outside $P$ from the hull. Together with step 1.1, this identifies ${\hat{T}}_{D^{2}}$ exactly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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, §1.2 and §2.6 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)