Source and significance assessment

Author: Henry Zweiman. Audit date: September 8, 2026.

Exact literature target

G. Kiss, G. Somlai, and T. Terpai, Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication, Journal of Algebra 664 (2025), 511–533, DOI 10.1016/j.jalgebra.2024.10.044.

The published final article, Section 7, pp. 530–531, asks for countably infinite invariant partitions of Q(alpha)'s positive elements and also discusses the polynomial ring and separation of positive affine generators. This manuscript proves that existence statement. The separate finite-class conjecture is not solved by this construction.

The source's finite constructions and two-class classification do not supply the endpoint partition. The primary comparison is the final published article, not an inference from its abstract or the older arXiv version.

Checked dependencies

The direct proof uses only rational-function zeros and poles, positivity on an interval, and one-sided limits. Its curve extension uses the smooth projective model of a one-variable function field in characteristic zero. The relevant Stacks Project Section 53.2, specifically Theorem 53.2.6 and Lemma 53.2.8, was inspected. Genericity of the real point, nonvanishing of a local differential, and countable infinitude are proved or explained in the manuscript rather than inferred from a drawing.

Search boundary

Searches inspected the final article, author manuscript, arXiv record, publisher record, and a current institutional author publication list. Queries combined the article title/authors with countable decomposition, semiring partitions, rational functions, positivity intervals, zeros, and poles. No matching later solution was found among the inspected results. This is bounded search evidence, not a guarantee against an unindexed preprint, an unpublished argument, or an overlooked result.

Contribution and limits

Unlike an ad hoc restricted case, the first theorem answers an explicit published question in full. The same proof mechanism extends through finite algebraic extensions of any genus and yields the finitely generated real-field theorem. Those are reasons to consider this a useful research note. The proof is short and elementary in the rational case; difficulty is not claimed from its length. Future citation counts and journal-level significance cannot be certified.

The manuscript is admitted to the public collection as one internally checked research manuscript with a proposed complete solution to a literature-posed problem. It is not independent peer review, and it does not retrospectively certify the older collection. No split into several papers is justified by its corollaries.