{"_id":"20249","isi":1,"article_type":"original","scopus_import":"1","publication_status":"published","oa":1,"OA_type":"hybrid","title":"Integral points on cubic surfaces: heuristics and numerics","external_id":{"arxiv":["2407.16315"],"isi":["001552779800001"]},"file_date_updated":"2025-09-03T06:44:44Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public","quality_controlled":"1","language":[{"iso":"eng"}],"type":"journal_article","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","orcid":"0000-0002-8314-0177","last_name":"Browning","full_name":"Browning, Timothy D"},{"last_name":"Wilsch","full_name":"Wilsch, Florian Alexander","first_name":"Florian Alexander","id":"560601DA-8D36-11E9-A136-7AC1E5697425","orcid":"0000-0001-7302-8256"}],"article_processing_charge":"Yes (via OA deal)","day":"01","project":[{"name":"New frontiers of the Manin conjecture","grant_number":"P32428","call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425"},{"grant_number":"P36278","name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"publication_identifier":{"issn":["1022-1824"],"eissn":["1420-9020"]},"citation":{"short":"T.D. Browning, F.A. Wilsch, Selecta Mathematica New Series 31 (2025).","ista":"Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 31(4), 81.","ama":"Browning TD, Wilsch FA. Integral points on cubic surfaces: heuristics and numerics. <i>Selecta Mathematica New Series</i>. 2025;31(4). doi:<a href=\"https://doi.org/10.1007/s00029-025-01074-1\">10.1007/s00029-025-01074-1</a>","chicago":"Browning, Timothy D, and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00029-025-01074-1\">https://doi.org/10.1007/s00029-025-01074-1</a>.","apa":"Browning, T. D., &#38; Wilsch, F. A. (2025). Integral points on cubic surfaces: heuristics and numerics. <i>Selecta Mathematica New Series</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00029-025-01074-1\">https://doi.org/10.1007/s00029-025-01074-1</a>","ieee":"T. D. Browning and F. A. Wilsch, “Integral points on cubic surfaces: heuristics and numerics,” <i>Selecta Mathematica New Series</i>, vol. 31, no. 4. Springer Nature, 2025.","mla":"Browning, Timothy D., and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” <i>Selecta Mathematica New Series</i>, vol. 31, no. 4, 81, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00029-025-01074-1\">10.1007/s00029-025-01074-1</a>."},"date_published":"2025-09-01T00:00:00Z","ddc":["500"],"license":"https://creativecommons.org/licenses/by/4.0/","department":[{"_id":"TiBr"}],"doi":"10.1007/s00029-025-01074-1","date_updated":"2025-09-30T14:29:25Z","date_created":"2025-08-31T22:01:31Z","article_number":"81","year":"2025","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"file":[{"checksum":"89352f1f7e8d2b367ae5f4e9bf9eb1f5","file_name":"2025_SelectaMathematica_Browning.pdf","creator":"dernst","date_created":"2025-09-03T06:44:44Z","success":1,"content_type":"application/pdf","relation":"main_file","date_updated":"2025-09-03T06:44:44Z","access_level":"open_access","file_id":"20281","file_size":2484757}],"abstract":[{"text":"We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.","lang":"eng"}],"publisher":"Springer Nature","intvolume":"        31","issue":"4","publication":"Selecta Mathematica New Series","volume":31,"has_accepted_license":"1","OA_place":"publisher","oa_version":"Published Version","arxiv":1,"corr_author":"1","month":"09","PlanS_conform":"1","acknowledgement":"The authors owe a debt of thanks to Yonatan Harpaz for asking about circle method heuristics for log K3 surfaces. His contribution to the resulting discussion is gratefully acknowledged. Thanks are also due to Andrew Sutherland for help with numerical data for the equation x^3 + y^3 + z^3 = 1, together with Alex Gamburd, Amit Ghosh, Peter Sarnak and Matteo Verzobio for their interest in this paper. Special thanks are due to Victor Wang for helpful conversations about the circle method heuristics and to the anonymous referee for several useful comments. While working on this paper, the authors were supported by a FWF grant (DOI 10.55776/P32428), and the first author was supported by a further FWF grant (DOI 10.55776/P36278) and a grant from the School of Mathematics at the Institute for Advanced Study in Princeton.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria)."}