{"paper":{"title":"An absolute bound for generalized Diophantine tuples over polynomial rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chi Hoi Yip, Kin Ming Tsang","submitted_at":"2026-07-01T16:50:11Z","abstract_excerpt":"Let $\\mathbb F$ be an algebraically closed field of characteristic $0$. Let $k\\geq 2$ be an integer, and let $n\\in \\mathbb F[x]\\setminus\\{0\\}$. We study generalized Diophantine tuples $A\\subset \\mathbb F[x]$ with property $D_k(n)$, meaning that $ab+n$ is a $k$-th power in $\\mathbb F[x]$ for all distinct elements $a,b\\in A$. For $k\\ge18$, we prove that every such tuple satisfies $|A|\\le6$, except for the necessary exceptional family in which $n=s^2$ is a $k$-th power and $A\\subset s\\mathbb{F}$. This bound is absolute: it is independent of both $n$ and $\\operatorname{deg} n$. Our proof develops "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01165","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01165/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"}