{"paper":{"title":"Sums of squares on curves and surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Bart{\\l}omiej Bychawski, Bartosz G{\\l}owacki, Tomasz Kowalczyk","submitted_at":"2026-06-21T09:15:40Z","abstract_excerpt":"We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m<M$. We then show that the $2s$-Pythagoras number of real algebras of the form $\\mathbb{R}[x,y,\\sqrt{f_1},\\sqrt{f_2},\\dots, \\sqrt{f_n}]$ are infinite, under some mild assumptions on the polynomials $f_1,f_2,\\dots, f_n \\in \\mathbb{R}[x,y]$. We prove that all of the higher even Pythagoras numbers are finite for the ring of $0$-regulous functions on a $0$-regulous variety. We then show that the codimension of the bad set of order $2n$, for $n>1$, can be of codimension $2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22401","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/2606.22401/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"}