[{"abstract":[{"text":"An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.","lang":"eng"}],"type":"journal_article","file":[{"content_type":"application/pdf","file_size":379158,"creator":"tbrownin","file_name":"wliqun.pdf","access_level":"open_access","date_created":"2019-04-16T09:12:20Z","date_updated":"2020-07-14T12:47:27Z","checksum":"a63594a3a91b4ba6e2a1b78b0720b3d0","relation":"main_file","file_id":"6311"}],"oa_version":"Submitted Version","ddc":["512"],"status":"public","title":"Counting rational points on biquadratic hypersurfaces","intvolume":" 349","_id":"6310","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","day":"20","has_accepted_license":"1","article_processing_charge":"No","scopus_import":"1","date_published":"2019-06-20T00:00:00Z","page":"920-940","publication":"Advances in Mathematics","citation":{"apa":"Browning, T. D., & Hu, L. Q. (2019). Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2019.04.031","ieee":"T. D. Browning and L. Q. Hu, “Counting rational points on biquadratic hypersurfaces,” Advances in Mathematics, vol. 349. Elsevier, pp. 920–940, 2019.","ista":"Browning TD, Hu LQ. 2019. Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. 349, 920–940.","ama":"Browning TD, Hu LQ. Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. 2019;349:920-940. doi:10.1016/j.aim.2019.04.031","chicago":"Browning, Timothy D, and L.Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” Advances in Mathematics. Elsevier, 2019. https://doi.org/10.1016/j.aim.2019.04.031.","short":"T.D. Browning, L.Q. Hu, Advances in Mathematics 349 (2019) 920–940.","mla":"Browning, Timothy D., and L. Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” Advances in Mathematics, vol. 349, Elsevier, 2019, pp. 920–40, doi:10.1016/j.aim.2019.04.031."},"file_date_updated":"2020-07-14T12:47:27Z","date_created":"2019-04-16T09:13:25Z","date_updated":"2023-08-25T10:11:55Z","volume":349,"author":[{"orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","last_name":"Browning","first_name":"Timothy D","full_name":"Browning, Timothy D"},{"first_name":"L.Q.","last_name":"Hu","full_name":"Hu, L.Q."}],"publication_status":"published","department":[{"_id":"TiBr"}],"publisher":"Elsevier","year":"2019","month":"06","publication_identifier":{"issn":["00018708"],"eissn":["10902082"]},"language":[{"iso":"eng"}],"doi":"10.1016/j.aim.2019.04.031","quality_controlled":"1","isi":1,"oa":1,"external_id":{"arxiv":["1810.08426"],"isi":["000468857300025"]}},{"volume":308,"date_updated":"2023-09-20T11:21:27Z","date_created":"2018-12-11T11:50:34Z","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Bárány, Imre","last_name":"Bárány","first_name":"Imre"},{"full_name":"Robins, Sinai","last_name":"Robins","first_name":"Sinai"}],"publisher":"Academic Press","department":[{"_id":"HeEd"}],"publication_status":"published","year":"2017","ec_funded":1,"publist_id":"6173","language":[{"iso":"eng"}],"doi":"10.1016/j.aim.2016.12.026","project":[{"name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734"}],"isi":1,"quality_controlled":"1","external_id":{"isi":["000409292900015"]},"main_file_link":[{"url":"https://arxiv.org/abs/1508.07594","open_access":"1"}],"oa":1,"publication_identifier":{"issn":["00018708"]},"month":"02","oa_version":"Submitted Version","intvolume":" 308","status":"public","title":"Algebraic vertices of non-convex polyhedra","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1180","abstract":[{"lang":"eng","text":"In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform."}],"type":"journal_article","date_published":"2017-02-21T00:00:00Z","page":"627 - 644","citation":{"short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics, vol. 308, Academic Press, 2017, pp. 627–44, doi:10.1016/j.aim.2016.12.026.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2016.12.026.","ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 2017;308:627-644. doi:10.1016/j.aim.2016.12.026","apa":"Akopyan, A., Bárány, I., & Robins, S. (2017). Algebraic vertices of non-convex polyhedra. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2016.12.026","ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” Advances in Mathematics, vol. 308. Academic Press, pp. 627–644, 2017.","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644."},"publication":"Advances in Mathematics","article_processing_charge":"No","day":"21","scopus_import":"1"}]