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