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