{"article_number":"12","date_published":"2025-09-01T00:00:00Z","title":"Counting integer points on affine surfaces with a side condition","external_id":{"arxiv":["2408.11453"]},"ec_funded":1,"oa_version":"Published Version","article_type":"original","month":"09","article_processing_charge":"No","author":[{"orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning"},{"orcid":"0000-0002-0854-0306","full_name":"Verzobio, Matteo","first_name":"Matteo","last_name":"Verzobio","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"corr_author":"1","ddc":["510"],"file_date_updated":"2026-02-12T07:50:47Z","citation":{"apa":"Browning, T. D., &#38; Verzobio, M. (2025). Counting integer points on affine surfaces with a side condition. <i>Discrete Analysis</i>. Cambridge: Alliance of Diamond Open Access Journals. <a href=\"https://doi.org/10.19086/da.143787\">https://doi.org/10.19086/da.143787</a>","short":"T.D. Browning, M. Verzobio, Discrete Analysis 2025 (2025).","mla":"Browning, Timothy D., and Matteo Verzobio. “Counting Integer Points on Affine Surfaces with a Side Condition.” <i>Discrete Analysis</i>, vol. 2025, 12, Cambridge: Alliance of Diamond Open Access Journals, 2025, doi:<a href=\"https://doi.org/10.19086/da.143787\">10.19086/da.143787</a>.","ieee":"T. D. Browning and M. Verzobio, “Counting integer points on affine surfaces with a side condition,” <i>Discrete Analysis</i>, vol. 2025. Cambridge: Alliance of Diamond Open Access Journals, 2025.","ama":"Browning TD, Verzobio M. Counting integer points on affine surfaces with a side condition. <i>Discrete Analysis</i>. 2025;2025. doi:<a href=\"https://doi.org/10.19086/da.143787\">10.19086/da.143787</a>","ista":"Browning TD, Verzobio M. 2025. Counting integer points on affine surfaces with a side condition. Discrete Analysis. 2025, 12.","chicago":"Browning, Timothy D, and Matteo Verzobio. “Counting Integer Points on Affine Surfaces with a Side Condition.” <i>Discrete Analysis</i>. Cambridge: Alliance of Diamond Open Access Journals, 2025. <a href=\"https://doi.org/10.19086/da.143787\">https://doi.org/10.19086/da.143787</a>."},"abstract":[{"text":"We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.","lang":"eng"}],"language":[{"iso":"eng"}],"doi":"10.19086/da.143787","acknowledgement":"Supported by FWF grant (DOI 10.55776/P36278), Supported by European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant\r\nAgreement No. 101034413.","publication_identifier":{"eissn":["2397-3129"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_type":"diamond","publication_status":"published","_id":"21003","file":[{"file_size":393625,"access_level":"open_access","creator":"dernst","checksum":"3d38e850b40f3e1abbfd30073bd4388a","date_created":"2026-02-12T07:50:47Z","success":1,"relation":"main_file","file_id":"21214","content_type":"application/pdf","date_updated":"2026-02-12T07:50:47Z","file_name":"2025_DiscreteAnalysis_Browning.pdf"}],"date_created":"2026-01-18T23:02:44Z","publication":"Discrete Analysis","publisher":"Cambridge: Alliance of Diamond Open Access Journals","scopus_import":"1","project":[{"grant_number":"P36278","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020"}],"date_updated":"2026-02-12T08:03:12Z","volume":2025,"has_accepted_license":"1","status":"public","arxiv":1,"OA_place":"publisher","day":"01","department":[{"_id":"TiBr"}],"intvolume":"      2025","type":"journal_article","license":"https://creativecommons.org/licenses/by/4.0/","oa":1}