DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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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Formal exponential, logarithm, and binomial powers over a commutative $Q$ -algebra

Definition

A commutative $Q$ -algebra here means a commutative ring $R$ equipped with a unital ring homomorphism $Q \to R$ . We identify each rational with its image in $R$ .

For $u \in x R ⟦ x ⟧$ , define

$\exp (u) : = \sum_{n \geq 0} \frac{u^{n}}{n!}, \log (1 + u) : = \sum_{n \geq 1} (- 1)^{n - 1} \frac{u^{n}}{n} .$

For $c \in R$ , define the formal binomial power

$(1 + u)^{c} : = \exp (c \log (1 + u)) .$

Since ${ord}_{x} (u^{n}) \geq n$ , each displayed family is summable. The unit criterion makes $1 + u$ a unit. These symbols name formal series only; no analytic exponential, logarithm, branch, or convergence is involved.

Depends on

Used by

Formal exp and log are inverse homomorphisms and formal binomial powers obey the expected addition laws Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

Formal exponential, logarithm, and binomial powers over a commutative QQ-algebra