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Fundamental arithmetic and atomic aspects of the additive structure of $\\mathbb{N}_0[\\alpha]$ were first studied by the second author and Correa-Morris (2022). In this paper, we continue the investigation, now from the valuation-theoretic perspective.\n  We show that for any algebraic number $\\alpha$, the additive monoid of $\\ma"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.02874","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-07-03T02:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"3c71da89900650b21f8e78c8ffc35b0fb14f9b2c382c89f784bbc99da571d65b","abstract_canon_sha256":"e72ec374665305a2f5a3ff8974e716d55b37c146101b6a53b1046c56aaec9e82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T01:16:35.048739Z","signature_b64":"5dKMyf7kSg/Vr8YTYk3QcR6iEk/9AtodEumAPBwq1tZkHTVxQdgQOB2ArMTrmlZfzzwVN62Ovoq5uGmTbHszCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"207bbaf72806029b6ee79b484e6e30522d7109cde755d3377201cf6216d2eba5","last_reissued_at":"2026-07-07T01:16:35.048167Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T01:16:35.048167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the additive structure of algebraic valuations of polynomial semirings II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alan Yao, Felix Gotti, Timothy Chen, Tony Lu","submitted_at":"2026-07-03T02:20:56Z","abstract_excerpt":"For $\\alpha \\in \\mathbb{C}$, let $\\mathbb{N}_0[\\alpha]$ be the subsemiring of~$\\mathbb{C}$ obtained as a homomorphic image of the $\\alpha$-evaluation map $\\mathbb{N}_0[x] \\to \\mathbb{C}$ defined as $p(x) \\mapsto p(\\alpha)$ for each polynomial $p(x) \\in \\mathbb{N}_0[x]$. Fundamental arithmetic and atomic aspects of the additive structure of $\\mathbb{N}_0[\\alpha]$ were first studied by the second author and Correa-Morris (2022). 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