{"paper":{"title":"Asymptotic Plateaus for Generalized Abel Equations with Financial Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generalized Abel differential equations with any polynomial degree n greater than or equal to 1 possess regular solutions that exhibit sharp growth rates and exact asymptotic plateaus on bounded and unbounded domains.","cross_cats":["cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Dragos-Patru Covei","submitted_at":"2026-05-04T17:07:25Z","abstract_excerpt":"We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\\prime }(x)=a_n(x)\\bigl(% y^n+\\lambda_{n-1}(x)y^{n-1}+\\dots+\\lambda_0(x)\\bigr)$ of arbitrary degree $% n\\ge1$ on the unbounded interval $[x_0,\\infty)$. Under mild structural hypotheses on the coefficients and on the existence of a stable moving equilibrium branch $E(x)$, we prove a new \\emph{Asymptotic Plateau Theorem} establishing that the solution issued from $y(x_0)=0$ is globally defined, strictly monotone, trapped between zero and $E(x)$, and converges to a finite positi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a comprehensive investigation into a generalized class of Abel ordinary differential equations (ODEs), extending the classical cubic form to arbitrary polynomial nonlinearities of degree n ≥ 1. [...] establishing the first systematic treatment of such generalizations in the literature.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Utilizing a unified barrier-based approach, we derive sharp growth rates and prove the existence of exact asymptotic plateaus, assuming this barrier method applies uniformly across all polynomial degrees n and both bounded and unbounded domains without additional restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Abel ODEs of arbitrary polynomial degree have existence, uniqueness, and sharp asymptotics proven via barriers, validated by Radau IIA numerics, and applied to financial modeling.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Abel differential equations with any polynomial degree n greater than or equal to 1 possess regular solutions that exhibit sharp growth rates and exact asymptotic plateaus on bounded and unbounded domains.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"95b926f2d43ff3e89e2ce2d7d34241aa8336ce6d8c3b0f92883c141270a127d9"},"source":{"id":"2605.02831","kind":"arxiv","version":2},"verdict":{"id":"47852f2a-d8d4-41b3-9012-7203a3b05fe1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T18:07:07.900739Z","strongest_claim":"We present a comprehensive investigation into a generalized class of Abel ordinary differential equations (ODEs), extending the classical cubic form to arbitrary polynomial nonlinearities of degree n ≥ 1. [...] establishing the first systematic treatment of such generalizations in the literature.","one_line_summary":"Generalized Abel ODEs of arbitrary polynomial degree have existence, uniqueness, and sharp asymptotics proven via barriers, validated by Radau IIA numerics, and applied to financial modeling.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Utilizing a unified barrier-based approach, we derive sharp growth rates and prove the existence of exact asymptotic plateaus, assuming this barrier method applies uniformly across all polynomial degrees n and both bounded and unbounded domains without additional restrictions.","pith_extraction_headline":"Generalized Abel differential equations with any polynomial degree n greater than or equal to 1 possess regular solutions that exhibit sharp growth rates and exact asymptotic plateaus on bounded and unbounded domains."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.02831/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T15:56:31.760081Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"842d275a9477d96098160d21632152d62a6d2714fa5879984f4250398779d926"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"8b8e1e9dd3e7afc14be0efd596ececd17438f38cccd9a17d79639efc9f1dca14"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}