PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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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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Rational formal power series are closed under sums and Cauchy products

Statement

Let $R$ be a commutative ring. If $F, G \in R ⟦ x ⟧$ are rational, then both $F + G$ and the Cauchy product $F G$ are rational. This includes zero series and polynomial presentations.

Facts & Assumptions

Given: Rational presentations $F = P_{1} / Q_{1}$ and $G = P_{2} / Q_{2}$ over a commutative ring $R$ .

[L1]

A rational formal series has a presentation $P / Q$ whose denominator has unit constant coefficient (Rational formal power series, proper presentations and reduced denominators).

[L2]

A formal series is a unit exactly when its constant coefficient is a unit (A formal power series is a unit exactly when its constant coefficient is a unit).

Proof

technique · direct

1.1givenL1L2

The product $Q_{1} Q_{2}$ has unit constant coefficient $Q_{1} (0) Q_{2} (0)$ , so it is a unit of $R ⟦ x ⟧$ by [L2].

2.1step 1.1L1algebra∎

The identities $F + G = (P_{1} Q_{2} + P_{2} Q_{1}) / (Q_{1} Q_{2})$ and $F G = P_{1} P_{2} / (Q_{1} Q_{2})$ have polynomial numerators and the denominator from step 1.1, so [L1] makes both series rational. The formulas remain valid when a numerator is zero or a denominator is $1$ .

Depends on

Used by

Words over a finite alphabet avoiding finitely many nonempty factors have a rational length generating function Theorem

Dependency tree · two levels

7 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

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Section 4.2 (standard reference, not scraped)