[{"volume":2017,"type":"journal_article","date_published":"2017-10-01T00:00:00Z","external_id":{"arxiv":["1402.4489"]},"article_type":"original","oa_version":"Preprint","publication":"Journal fur die Reine und Angewandte Mathematik","doi":"10.1515/crelle-2014-0122","date_updated":"2024-03-05T12:09:21Z","publisher":"Walter de Gruyter","article_processing_charge":"No","extern":"1","quality_controlled":"1","oa":1,"month":"10","page":"122","author":[{"full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning","orcid":"0000-0002-8314-0177"},{"last_name":"Prendiville","full_name":"Prendiville, Sean","first_name":"Sean"}],"publication_status":"published","publist_id":"7646","day":"01","arxiv":1,"date_created":"2018-12-11T11:45:28Z","year":"2017","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"271","relation":"earlier_version","status":"public"}]},"status":"public","main_file_link":[{"url":"https://arxiv.org/abs/1402.4489","open_access":"1"}],"intvolume":"      2017","citation":{"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>","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.","short":"T.D. Browning, S. Prendiville, Journal Fur Die Reine Und Angewandte Mathematik 2017 (2017) 122.","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.","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>.","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>."},"_id":"256","title":"Improvements in Birch's theorem on forms in many variables","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"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"While working on this paper the authors were supported by the Leverhulme Trust and ERC grant 306457.","issue":"731","publication_identifier":{"issn":["0075-4102"]}},{"issue":"731","publication_identifier":{"issn":["0075-4102"]},"status":"public","intvolume":"      2017","citation":{"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>.","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>.","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>","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.","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>","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."},"main_file_link":[{"url":"https://arxiv.org/abs/1402.4489","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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"}],"title":"Improvements in Birch's theorem on forms in many variables","_id":"271","acknowledgement":"While working on this paper the authors were supported by the Leverhulme Trust and ERC grant 306457.","author":[{"full_name":"Browning, Timothy D","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","orcid":"0000-0002-8314-0177"},{"last_name":"Prendiville","first_name":"Sean","full_name":"Prendiville, Sean"}],"page":"203 - 234","date_created":"2018-12-11T11:45:32Z","arxiv":1,"day":"20","publication_status":"submitted","publist_id":"7631","related_material":{"record":[{"id":"256","relation":"later_version","status":"public"}]},"language":[{"iso":"eng"}],"year":"2015","date_published":"2015-02-20T00:00:00Z","type":"journal_article","volume":2017,"oa_version":"Preprint","article_type":"original","external_id":{"arxiv":["1402.4489"]},"article_processing_charge":"No","publisher":"Walter de Gruyter","date_updated":"2024-03-05T12:09:22Z","doi":"10.1515/crelle-2014-0122","publication":"Journal fur die Reine und Angewandte Mathematik","month":"02","oa":1,"quality_controlled":"1","extern":"1"}]