[{"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2212.10373","open_access":"1"}],"month":"08","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-09-06T22:01:58Z","OA_type":"green","arxiv":1,"oa":1,"day":"25","_id":"22821","type":"journal_article","quality_controlled":"1","corr_author":"1","article_type":"original","language":[{"iso":"eng"}],"article_processing_charge":"No","date_published":"2026-08-25T00:00:00Z","status":"public","publication_status":"epub_ahead","abstract":[{"lang":"eng","text":"With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree 𝑑 polynomials 𝑓∈ℤ⁢[𝑡] that are ordered by height. This allows us to establish averaged versions of the Bateman–Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of 𝑓, both with arbitrary logarithmic power savings."}],"oa_version":"Preprint","publication_identifier":{"issn":["0075-4102"],"eissn":["1435-5345"]},"acknowledgement":"Tim Browning was supported by an FWF grant (DOI 10.55776/P32428), Efthymios Sofos was supported by EPSRC New Horizons grant EP/V048236/1, and Joni Teräväinen was supported by Academy of Finland grant no. 340098, a von Neumann Fellowship (NSF grant DMS-1926686), and funding from European Union’s Horizon Europe research and innovation programme under Marie Skłodowska-Curie grant agreement no. 101058904. This material is based upon work supported by a grant from the Institute for Advanced Study School of Mathematics. The authors are extremely grateful to the anonymous referee and to Yijie Diao for useful comments.","fulldoi":"https://doi.org/10.1515/crelle-2026-0062","project":[{"grant_number":"P32428","name":"New frontiers of the Manin conjecture","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"publication":"Journal für die reine und angewandte Mathematik","author":[{"full_name":"Browning, Timothy D","first_name":"Timothy D","orcid":"0000-0002-8314-0177","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Sofos","full_name":"Sofos, Efthymios","first_name":"Efthymios"},{"first_name":"Joni","full_name":"Teräväinen, Joni","last_name":"Teräväinen"}],"title":"Bateman-Horn, polynomial Chowla and the Hasse principle with probability","year":"2026","citation":{"ista":"Browning TD, Sofos E, Teräväinen J. 2026. Bateman-Horn, polynomial Chowla and the Hasse principle with probability. Journal für die reine und angewandte Mathematik.","ama":"Browning TD, Sofos E, Teräväinen J. Bateman-Horn, polynomial Chowla and the Hasse principle with probability. <i>Journal für die reine und angewandte Mathematik</i>. 2026. doi:<a href=\"https://doi.org/10.1515/crelle-2026-0062\">10.1515/crelle-2026-0062</a>","chicago":"Browning, Timothy D, Efthymios Sofos, and Joni Teräväinen. “Bateman-Horn, Polynomial Chowla and the Hasse Principle with Probability.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>. De Gruyter, 2026. <a href=\"https://doi.org/10.1515/crelle-2026-0062\">https://doi.org/10.1515/crelle-2026-0062</a>.","ieee":"T. D. Browning, E. Sofos, and J. Teräväinen, “Bateman-Horn, polynomial Chowla and the Hasse principle with probability,” <i>Journal für die reine und angewandte Mathematik</i>. De Gruyter, 2026.","apa":"Browning, T. D., Sofos, E., &#38; Teräväinen, J. (2026). Bateman-Horn, polynomial Chowla and the Hasse principle with probability. <i>Journal Für Die Reine Und Angewandte Mathematik</i>. De Gruyter. <a href=\"https://doi.org/10.1515/crelle-2026-0062\">https://doi.org/10.1515/crelle-2026-0062</a>","short":"T.D. Browning, E. Sofos, J. Teräväinen, Journal Für Die Reine Und Angewandte Mathematik (2026).","mla":"Browning, Timothy D., et al. “Bateman-Horn, Polynomial Chowla and the Hasse Principle with Probability.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>, De Gruyter, 2026, doi:<a href=\"https://doi.org/10.1515/crelle-2026-0062\">10.1515/crelle-2026-0062</a>."},"department":[{"_id":"TiBr"}],"publisher":"De Gruyter","OA_place":"repository","external_id":{"arxiv":["2212.10373"]},"doi":"10.1515/crelle-2026-0062","date_updated":"2026-09-09T13:28:45Z","scopus_import":"1"},{"oa":1,"quality_controlled":"1","corr_author":"1","day":"01","_id":"256","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:45:28Z","month":"10","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1402.4489"}],"intvolume":"      2017","arxiv":1,"publication_status":"published","abstract":[{"text":"We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over ℝ and over ℚp for all primes p, provided that the form has at least (d - 1/2 √d)2d variables. This improves on a longstanding result of Birch.","lang":"eng"}],"status":"public","date_published":"2017-10-01T00:00:00Z","article_processing_charge":"No","oa_version":"Preprint","extern":"1","publist_id":"7646","article_type":"original","language":[{"iso":"eng"}],"publication":"Journal fur die Reine und Angewandte Mathematik","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","first_name":"Timothy D"},{"last_name":"Prendiville","full_name":"Prendiville, Sean","first_name":"Sean"}],"fulldoi":"https://doi.org/10.1515/crelle-2014-0122","title":"Improvements in Birch's theorem on forms in many variables","page":"122","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"271"}]},"publication_identifier":{"issn":["0075-4102"]},"acknowledgement":"While working on this paper the authors were supported by the Leverhulme Trust and ERC grant 306457.","issue":"731","doi":"10.1515/crelle-2014-0122","date_updated":"2024-10-09T20:58:16Z","external_id":{"arxiv":["1402.4489"]},"publisher":"Walter de Gruyter","volume":2017,"citation":{"ieee":"T. D. Browning and S. Prendiville, “Improvements in Birch’s theorem on forms in many variables,” <i>Journal fur die Reine und Angewandte Mathematik</i>, vol. 2017, no. 731. Walter de Gruyter, p. 122, 2017.","ista":"Browning TD, Prendiville S. 2017. Improvements in Birch’s theorem on forms in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731), 122.","ama":"Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. <i>Journal fur die Reine und Angewandte Mathematik</i>. 2017;2017(731):122. doi:<a href=\"https://doi.org/10.1515/crelle-2014-0122\">10.1515/crelle-2014-0122</a>","chicago":"Browning, Timothy D, and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter, 2017. <a href=\"https://doi.org/10.1515/crelle-2014-0122\">https://doi.org/10.1515/crelle-2014-0122</a>.","short":"T.D. Browning, S. Prendiville, Journal Fur Die Reine Und Angewandte Mathematik 2017 (2017) 122.","apa":"Browning, T. D., &#38; Prendiville, S. (2017). Improvements in Birch’s theorem on forms in many variables. <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter. <a href=\"https://doi.org/10.1515/crelle-2014-0122\">https://doi.org/10.1515/crelle-2014-0122</a>","mla":"Browning, Timothy D., and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>, vol. 2017, no. 731, Walter de Gruyter, 2017, p. 122, doi:<a href=\"https://doi.org/10.1515/crelle-2014-0122\">10.1515/crelle-2014-0122</a>."},"year":"2017"},{"extern":"1","oa_version":"Preprint","article_processing_charge":"No","date_published":"2015-02-20T00:00:00Z","abstract":[{"text":"We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the form has at least (d-\\sqrt{d}/2)2^d variables. This improves on a longstanding result of Birch.","lang":"eng"}],"publication_status":"submitted","status":"public","language":[{"iso":"eng"}],"article_type":"original","publist_id":"7631","_id":"271","day":"20","type":"journal_article","quality_controlled":"1","corr_author":"1","oa":1,"arxiv":1,"intvolume":"      2017","main_file_link":[{"url":"https://arxiv.org/abs/1402.4489","open_access":"1"}],"month":"02","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:45:32Z","external_id":{"arxiv":["1402.4489"]},"doi":"10.1515/crelle-2014-0122","date_updated":"2024-10-09T20:58:15Z","year":"2015","citation":{"ieee":"T. D. Browning and S. Prendiville, “Improvements in Birch’s theorem on forms in many variables,” <i>Journal fur die Reine und Angewandte Mathematik</i>, vol. 2017, no. 731. Walter de Gruyter, pp. 203–234.","ista":"Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731), 203–234.","ama":"Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. <i>Journal fur die Reine und Angewandte Mathematik</i>. 2017(731):203-234. doi:<a href=\"https://doi.org/10.1515/crelle-2014-0122\">10.1515/crelle-2014-0122</a>","chicago":"Browning, Timothy D, and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter, n.d. <a href=\"https://doi.org/10.1515/crelle-2014-0122\">https://doi.org/10.1515/crelle-2014-0122</a>.","short":"T.D. Browning, S. Prendiville, Journal Fur Die Reine Und Angewandte Mathematik 2017 (n.d.) 203–234.","apa":"Browning, T. D., &#38; Prendiville, S. (n.d.). Improvements in Birch’s theorem on forms in many variables. <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter. <a href=\"https://doi.org/10.1515/crelle-2014-0122\">https://doi.org/10.1515/crelle-2014-0122</a>","mla":"Browning, Timothy D., and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>, vol. 2017, no. 731, Walter de Gruyter, pp. 203–34, doi:<a href=\"https://doi.org/10.1515/crelle-2014-0122\">10.1515/crelle-2014-0122</a>."},"volume":2017,"publisher":"Walter de Gruyter","page":"203 - 234","title":"Improvements in Birch's theorem on forms in many variables","fulldoi":"https://doi.org/10.1515/crelle-2014-0122","publication":"Journal fur die Reine und Angewandte Mathematik","author":[{"last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","first_name":"Timothy D","orcid":"0000-0002-8314-0177"},{"first_name":"Sean","full_name":"Prendiville, Sean","last_name":"Prendiville"}],"issue":"731","acknowledgement":"While working on this paper the authors were supported by the Leverhulme Trust and ERC grant 306457.","publication_identifier":{"issn":["0075-4102"]},"related_material":{"record":[{"relation":"later_version","status":"public","id":"256"}]}},{"OA_type":"green","arxiv":1,"month":"11","main_file_link":[{"url":"https://arxiv.org/abs/0707.2296","open_access":"1"}],"date_created":"2018-12-11T11:45:14Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"212","day":"26","type":"journal_article","oa":1,"publist_id":"7701","article_type":"original","language":[{"iso":"eng"}],"extern":"1","oa_version":"None","article_processing_charge":"No","abstract":[{"lang":"eng","text":"For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε &gt; 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε."}],"publication_status":"published","status":"public","date_published":"2005-11-26T00:00:00Z","issue":"584","publication_identifier":{"issn":["0075-4102"]},"title":"Counting rational points on hypersurfaces","page":"83 - 115","fulldoi":"https://doi.org/https://doi.org/10.1515/crll.2005.2005.584.83","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","full_name":"Browning, Timothy D","first_name":"Timothy D","orcid":"0000-0002-8314-0177"},{"full_name":"Heath Brown, Roger","first_name":"Roger","last_name":"Heath Brown"}],"publication":"Journal fur die Reine und Angewandte Mathematik","OA_place":"repository","citation":{"chicago":"Browning, Timothy D, and Roger Heath Brown. “Counting Rational Points on Hypersurfaces.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter and Co , 2005. <a href=\"https://doi.org/10.1515/crll.2005.2005.584.83\">https://doi.org/10.1515/crll.2005.2005.584.83</a>.","ama":"Browning TD, Heath Brown R. Counting rational points on hypersurfaces. <i>Journal fur die Reine und Angewandte Mathematik</i>. 2005;(584):83-115. doi:<a href=\"https://doi.org/10.1515/crll.2005.2005.584.83\">https://doi.org/10.1515/crll.2005.2005.584.83</a>","ista":"Browning TD, Heath Brown R. 2005. Counting rational points on hypersurfaces. Journal fur die Reine und Angewandte Mathematik. (584), 83–115.","ieee":"T. D. Browning and R. Heath Brown, “Counting rational points on hypersurfaces,” <i>Journal fur die Reine und Angewandte Mathematik</i>, no. 584. Walter de Gruyter and Co , pp. 83–115, 2005.","mla":"Browning, Timothy D., and Roger Heath Brown. “Counting Rational Points on Hypersurfaces.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>, no. 584, Walter de Gruyter and Co , 2005, pp. 83–115, doi:<a href=\"https://doi.org/10.1515/crll.2005.2005.584.83\">https://doi.org/10.1515/crll.2005.2005.584.83</a>.","short":"T.D. Browning, R. Heath Brown, Journal Fur Die Reine Und Angewandte Mathematik (2005) 83–115.","apa":"Browning, T. D., &#38; Heath Brown, R. (2005). Counting rational points on hypersurfaces. <i>Journal Fur Die Reine Und Angewandte Mathematik</i>. Walter de Gruyter and Co . <a href=\"https://doi.org/10.1515/crll.2005.2005.584.83\">https://doi.org/10.1515/crll.2005.2005.584.83</a>"},"year":"2005","publisher":"Walter de Gruyter and Co ","date_updated":"2026-08-21T09:31:08Z","doi":"https://doi.org/10.1515/crll.2005.2005.584.83","external_id":{"arxiv":["abs/math/0404456"]}}]