{"paper":{"title":"Values of algebraic functions at Liouville numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Algebraic functions of degree m map all L-numbers to U_m-numbers under general assumptions.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Diego Marques, Yuri Bilu","submitted_at":"2026-04-09T23:17:42Z","abstract_excerpt":"In 1953 LeVeque proved the existence of $U_m$-numbers by showing that for some specially defined Liouville number $\\lambda$, the $m$th root $\\lambda^{1/m}$ is in $U_m$. In this article we study the following question: let $u$ be an algebraic function of degree $m$ and $\\lambda$ a Liouville number; under which conditions is $u(\\lambda)$ a $U_m$-number? We consider a more refined notion of $\\mathcal{L}$-numbers, and show that, under very general assumptions, an algebraic function of degree $m$ takes $U_m$-values at all $\\mathcal{L}$-numbers."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under very general assumptions, an algebraic function of degree m takes U_m-values at all L-numbers.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The 'very general assumptions' on the algebraic function u of degree m and the precise definition of the class of L-numbers are broad enough for the claimed statement to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Algebraic functions of degree m take U_m-values at all L-numbers under general assumptions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Algebraic functions of degree m map all L-numbers to U_m-numbers under general assumptions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4756355401676c16b11239b69cec4aaff766f450a12b12f1f5bc4cd756f6c0ac"},"source":{"id":"2604.08818","kind":"arxiv","version":3},"verdict":{"id":"9bee4b58-eef7-4936-bc72-86ed4e306693","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:39:24.018491Z","strongest_claim":"under very general assumptions, an algebraic function of degree m takes U_m-values at all L-numbers.","one_line_summary":"Algebraic functions of degree m take U_m-values at all L-numbers under general assumptions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The 'very general assumptions' on the algebraic function u of degree m and the precise definition of the class of L-numbers are broad enough for the claimed statement to hold.","pith_extraction_headline":"Algebraic functions of degree m map all L-numbers to U_m-numbers under general assumptions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.08818/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"}