{"paper":{"title":"On rank in algebraic closure","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC","math.AG"],"primary_cat":"math.NT","authors_text":"Amichai Lampert, Tamar Ziegler","submitted_at":"2022-05-11T08:09:39Z","abstract_excerpt":"Let $ {\\mathbf k} $ be a field and $Q\\in {\\mathbf k}[x_1, \\ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\\mathbf k}$-Schmidt rank $rk_{\\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \\sum_{i=1}^r R_iS_i$ with $R_i, S_i \\in {\\mathbf k}[x_1, \\ldots, x_s]$ forms of degree $<d$. When $ {\\mathbf k} $ is algebraically closed, this rank is essentially equivalent to the codimension in $ {\\mathbf k}^s $ of the singular locus of the variety defined by $ Q, $ known also as the Birch rank of $ Q. $ When $ {\\mathbf k} $ is a number field, a finite field or a function field, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.05329","kind":"arxiv","version":3},"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/2205.05329/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"}