{"paper":{"title":"On the structure and generic non-Cartesianity of polynomials in product spaces","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Chun-Yen Shen, Tuyen Trung Truong, Wei-Hsuan Yu","submitted_at":"2026-05-21T11:06:30Z","abstract_excerpt":"We develop a general theory of Cartesian and non-Cartesian polynomials on products of complex spaces $\\mathbb{C}^{n_1} \\times \\cdots \\times \\mathbb{C}^{n_k}$. We prove that, for any fixed degree $d \\ge 2$, a (Zariski) generic polynomial is non-Cartesian in a broad range of dimensions, establishing that Cartesian structure is highly exceptional.\n  We further introduce effective sufficient criteria for a polynomial to be non-Cartesian. Moreover, we show that being (non)-Catersian can be decided algorithmically via Gr\\\"obner basis methods and quantitative forms of Hilbert's Nullstellensatz.\n  As "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22320","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/2605.22320/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"}