{"paper":{"title":"Rota Baxter Operators on Truncated Polynomial Algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Rota-Baxter operators of weights zero and one on the truncated polynomial algebra R = K[x1,…,xn]/m² are classified by nilpotency conditions or by idempotent endomorphisms on the maximal ideal quotient.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Azhar Farooq","submitted_at":"2026-05-15T06:46:50Z","abstract_excerpt":"Let K be a field of characteristic zero, and let m=(x_1,...,x_n)) be a maximal ideal of the polynomial ring K[x_1,...,x_n]. We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,\\dots,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P) \\subset m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)\\in{0,-1}. On the maximal ideal m/m^2, such "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,…,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P)⊂m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)∈{0,−1}. On the maximal ideal m/m^2, such operators induce an endomorphism L satisfying L^2+L=0, equivalently, −L is idempotent. We further show that each family is isomorphic to the variety of idempotent matrices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The claim that a standard rescaling reduces the nonzero-weight case to weight one without loss of generality, and that the resulting operators on m/m^2 always induce an endomorphism L with L^2 + L = 0, as stated in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Classifies Rota-Baxter operators on truncated polynomial algebras as maps satisfying P squared equals zero or inducing idempotent endomorphisms on the maximal ideal, with each family corresponding to the variety of idempotent matrices.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Rota-Baxter operators of weights zero and one on the truncated polynomial algebra R = K[x1,…,xn]/m² are classified by nilpotency conditions or by idempotent endomorphisms on the maximal ideal quotient.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e3357327bd617c0cf88427e3e373c955178b67c213e3593f9aafef3d624d728e"},"source":{"id":"2605.15670","kind":"arxiv","version":1},"verdict":{"id":"6d7c89b4-b2fb-4a73-abcb-18e3e8a55ae7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:14:08.435722Z","strongest_claim":"We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,…,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P)⊂m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)∈{0,−1}. On the maximal ideal m/m^2, such operators induce an endomorphism L satisfying L^2+L=0, equivalently, −L is idempotent. We further show that each family is isomorphic to the variety of idempotent matrices.","one_line_summary":"Classifies Rota-Baxter operators on truncated polynomial algebras as maps satisfying P squared equals zero or inducing idempotent endomorphisms on the maximal ideal, with each family corresponding to the variety of idempotent matrices.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The claim that a standard rescaling reduces the nonzero-weight case to weight one without loss of generality, and that the resulting operators on m/m^2 always induce an endomorphism L with L^2 + L = 0, as stated in the abstract.","pith_extraction_headline":"Rota-Baxter operators of weights zero and one on the truncated polynomial algebra R = K[x1,…,xn]/m² are classified by nilpotency conditions or by idempotent endomorphisms on the maximal ideal quotient."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15670/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:34.501972Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T18:31:18.800239Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:21:42.569306Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:56.064346Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"cce1d46c349ee91fcc79e688a1611250fd1de6e83db26c50a0bbeb2400e19202"},"references":{"count":16,"sample":[{"doi":"","year":1960,"title":"Baxter,An analytic problem whose solution follows from a simple algebraic identity, Pacific J","work_id":"42890494-b1f0-4465-be04-f97b300d089d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"Rota,Baxter algebras and combinatorial identities I, II, Bull","work_id":"ad8b4f0c-d221-4d93-80e7-940e6ce55033","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"A. 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