{"paper":{"title":"Finite Generation in Polynomial Semirings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Mohammad El Asal, Wael Mahboub","submitted_at":"2026-04-13T14:50:35Z","abstract_excerpt":"We study the semiring $\\mathbb{N}_0[\\alpha]$ as an additive monoid where $\\alpha$ is a positive real algebraic number. In the atomic case, the atoms of $\\mathbb{N}_0[\\alpha]$ are precisely the powers $\\alpha^n$ up to a certain nonnegative integer $n$, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form $\\mathfrak{m}_\\alpha(X)=p_\\alpha(X)-c$ with $c\\in\\mathbb{N}$. Our second main result shows that finite generation forces $\\alpha$ to be a weak P"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our second main result shows that finite generation forces alpha to be a weak Perron number.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes the atomic case where atoms are precisely the powers alpha^n up to a certain n, and relies on divisibility conditions involving negative-tail polynomials without independent verification of atomicity in all cases.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finite generation of the additive monoid N_0[alpha] is fully characterized for minimal polynomials of the form p(X) - c and implies that alpha must be a weak Perron number, with applications to cubic cases and rank-3 monoids.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"97f436a5c16a47ef6b33c71b52ee2140cf4e449690a3e76cf0280abc8f498331"},"source":{"id":"2604.11569","kind":"arxiv","version":2},"verdict":{"id":"447d9663-ac86-4ae5-96ba-26cd0745c89b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:07:24.982845Z","strongest_claim":"Our second main result shows that finite generation forces alpha to be a weak Perron number.","one_line_summary":"Finite generation of the additive monoid N_0[alpha] is fully characterized for minimal polynomials of the form p(X) - c and implies that alpha must be a weak Perron number, with applications to cubic cases and rank-3 monoids.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes the atomic case where atoms are precisely the powers alpha^n up to a certain n, and relies on divisibility conditions involving negative-tail polynomials without independent verification of atomicity in all cases.","pith_extraction_headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11569/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}