DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Dependency tree · 2 levels

Sources

Benjamin Sambale, An Invitation to Formal Power Series

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

Definition

Depends on

Dependency tree · 2 levels

Sources

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