{"paper":{"title":"On the Fermat-type partial differential-difference equations on $\\mathbb{C}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jun Wang, Tingbin Cao, Zhuan Ye","submitted_at":"2025-09-02T01:06:51Z","abstract_excerpt":"Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \\cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\\mathbb{C}^{n}.$ Let $ L(f)=\\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and $\\overline{f}(z)=f(z_{1}+c_{1}, \\ldots, z_{n}+c_{n})$, where $q_{t_j}$ ($j=1,2, \\cdots, s\\le n$) are non-zero polynomials on $\\mathbb{C}^{n}$ and $c=(c_{1}, \\ldots, c_{n})\\in \\mathbb{C}^{n}\\setminus\\{0\\}$. We show the structures of all entire solutions to the non-linear partial differential-difference equation $$(p_{1} L(f)+p_{2}\\overline{f}+p_5 f)^{2}+(p_{3}L(f)+p_{4}\\overline{f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01862","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/2509.01862/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"}