--- _id: '215' abstract: - lang: eng text: For any n≥3, let F ∈ Z[X0,...,Xn ] be a form of degree d *≥5 that defines a non-singular hypersurface X ⊂ Pn . The main result in this paper is a proof of the fact that the number N (F ; B) of Q-rational points on X which have height at most B satisfiesN (F ; B) = Od,ε,n (Bn −1+ε ), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n. New estimates are also obtained for the number of representations of a positive integer as the sum of three dth powers, and for the paucity of integer solutions to equal sums of like polynomials.* author: - first_name: Timothy D full_name: Timothy Browning id: 35827D50-F248-11E8-B48F-1D18A9856A87 last_name: Browning orcid: 0000-0002-8314-0177 - first_name: Roger full_name: Heath-Brown, Roger last_name: Heath Brown citation: ama: Browning TD, Heath Brown R. The density of rational points on non-singular hypersurfaces, I. Bulletin of the London Mathematical Society. 2006;38(3):401-410. doi:10.1112/S0024609305018412 apa: Browning, T. D., & Heath Brown, R. (2006). The density of rational points on non-singular hypersurfaces, I. Bulletin of the London Mathematical Society. Wiley-Blackwell. https://doi.org/10.1112/S0024609305018412 chicago: Browning, Timothy D, and Roger Heath Brown. “The Density of Rational Points on Non-Singular Hypersurfaces, I.” Bulletin of the London Mathematical Society. Wiley-Blackwell, 2006. https://doi.org/10.1112/S0024609305018412. ieee: T. D. Browning and R. Heath Brown, “The density of rational points on non-singular hypersurfaces, I,” Bulletin of the London Mathematical Society, vol. 38, no. 3. Wiley-Blackwell, pp. 401–410, 2006. ista: Browning TD, Heath Brown R. 2006. The density of rational points on non-singular hypersurfaces, I. Bulletin of the London Mathematical Society. 38(3), 401–410. mla: Browning, Timothy D., and Roger Heath Brown. “The Density of Rational Points on Non-Singular Hypersurfaces, I.” Bulletin of the London Mathematical Society, vol. 38, no. 3, Wiley-Blackwell, 2006, pp. 401–10, doi:10.1112/S0024609305018412. short: T.D. Browning, R. Heath Brown, Bulletin of the London Mathematical Society 38 (2006) 401–410. date_created: 2018-12-11T11:45:15Z date_published: 2006-12-23T00:00:00Z date_updated: 2021-01-12T06:55:36Z day: '23' doi: 10.1112/S0024609305018412 extern: 1 intvolume: ' 38' issue: '3' month: '12' page: 401 - 410 publication: Bulletin of the London Mathematical Society publication_status: published publisher: Wiley-Blackwell publist_id: '7697' quality_controlled: 0 status: public title: The density of rational points on non-singular hypersurfaces, I type: journal_article volume: 38 year: '2006' ... --- _id: '216' abstract: - lang: eng text: For any N ≥ 2, let Z ⊂ ℙN be a geometrically integral algebraic variety of degree d. This article is concerned with the number Nz(B) of ℚ-rational points on Z which have height at most B. For any ε > 0, we establish the estimate NZ(B) = O d,ε,N(Bdim Z+ε), provided that d ≥ 6. As indicated, the implied constant depends at most on d, ε, and N. author: - first_name: Timothy D full_name: Timothy Browning id: 35827D50-F248-11E8-B48F-1D18A9856A87 last_name: Browning orcid: 0000-0002-8314-0177 - first_name: Roger full_name: Heath-Brown, Roger last_name: Heath Brown - first_name: Per full_name: Salberger, Per last_name: Salberger citation: ama: Browning TD, Heath Brown R, Salberger P. Counting rational points on algebraic varieties. Duke Mathematical Journal. 2006;132(3):545-578. doi:10.1215/S0012-7094-06-13236-2 apa: Browning, T. D., Heath Brown, R., & Salberger, P. (2006). Counting rational points on algebraic varieties. Duke Mathematical Journal. Unknown. https://doi.org/10.1215/S0012-7094-06-13236-2 chicago: Browning, Timothy D, Roger Heath Brown, and Per Salberger. “Counting Rational Points on Algebraic Varieties.” Duke Mathematical Journal. Unknown, 2006. https://doi.org/10.1215/S0012-7094-06-13236-2. ieee: T. D. Browning, R. Heath Brown, and P. Salberger, “Counting rational points on algebraic varieties,” Duke Mathematical Journal, vol. 132, no. 3. Unknown, pp. 545–578, 2006. ista: Browning TD, Heath Brown R, Salberger P. 2006. Counting rational points on algebraic varieties. Duke Mathematical Journal. 132(3), 545–578. mla: Browning, Timothy D., et al. “Counting Rational Points on Algebraic Varieties.” Duke Mathematical Journal, vol. 132, no. 3, Unknown, 2006, pp. 545–78, doi:10.1215/S0012-7094-06-13236-2. short: T.D. Browning, R. Heath Brown, P. Salberger, Duke Mathematical Journal 132 (2006) 545–578. date_created: 2018-12-11T11:45:15Z date_published: 2006-04-15T00:00:00Z date_updated: 2021-01-12T06:55:41Z day: '15' doi: 10.1215/S0012-7094-06-13236-2 extern: 1 intvolume: ' 132' issue: '3' month: '04' page: 545 - 578 publication: Duke Mathematical Journal publication_status: published publisher: Unknown publist_id: '7696' quality_controlled: 0 status: public title: Counting rational points on algebraic varieties type: journal_article volume: 132 year: '2006' ... --- _id: '218' abstract: - lang: eng text: This paper is concerned with the average order of certain arithmetic functions, as they range over the values taken by binary forms. author: - first_name: Régis full_name: de la Bretèche, Régis last_name: De La Bretèche - first_name: Timothy D full_name: Timothy Browning id: 35827D50-F248-11E8-B48F-1D18A9856A87 last_name: Browning orcid: 0000-0002-8314-0177 citation: ama: De La Bretèche R, Browning TD. Sums of arithmetic functions over values of binary forms. Acta Arithmetica. 2006;125(3):291-304. doi:10.4064/aa125-3-6 apa: De La Bretèche, R., & Browning, T. D. (2006). Sums of arithmetic functions over values of binary forms. Acta Arithmetica. Instytut Matematyczny. https://doi.org/10.4064/aa125-3-6 chicago: De La Bretèche, Régis, and Timothy D Browning. “Sums of Arithmetic Functions over Values of Binary Forms.” Acta Arithmetica. Instytut Matematyczny, 2006. https://doi.org/10.4064/aa125-3-6. ieee: R. De La Bretèche and T. D. Browning, “Sums of arithmetic functions over values of binary forms,” Acta Arithmetica, vol. 125, no. 3. Instytut Matematyczny, pp. 291–304, 2006. ista: De La Bretèche R, Browning TD. 2006. Sums of arithmetic functions over values of binary forms. Acta Arithmetica. 125(3), 291–304. mla: De La Bretèche, Régis, and Timothy D. Browning. “Sums of Arithmetic Functions over Values of Binary Forms.” Acta Arithmetica, vol. 125, no. 3, Instytut Matematyczny, 2006, pp. 291–304, doi:10.4064/aa125-3-6. short: R. De La Bretèche, T.D. Browning, Acta Arithmetica 125 (2006) 291–304. date_created: 2018-12-11T11:45:16Z date_published: 2006-01-01T00:00:00Z date_updated: 2021-01-12T06:55:49Z day: '01' doi: 10.4064/aa125-3-6 extern: 1 intvolume: ' 125' issue: '3' month: '01' page: 291 - 304 publication: Acta Arithmetica publication_status: published publisher: Instytut Matematyczny publist_id: '7694' quality_controlled: 0 status: public title: Sums of arithmetic functions over values of binary forms type: journal_article volume: 125 year: '2006' ... --- _id: '2333' alternative_title: - Contemporary Mathematics author: - first_name: Élliott full_name: Lieb, Élliott H last_name: Lieb - first_name: Robert full_name: Robert Seiringer id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 - first_name: Jan full_name: Solovej, Jan P last_name: Solovej citation: ama: 'Lieb É, Seiringer R, Solovej J. Ground-state energy of a dilute Fermi gas. In: Vol 412. American Mathematical Society; 2006:239-248. doi:10.1090/conm/412' apa: Lieb, É., Seiringer, R., & Solovej, J. (2006). Ground-state energy of a dilute Fermi gas (Vol. 412, pp. 239–248). Presented at the Differential Equations and Mathematical Physics, American Mathematical Society. https://doi.org/10.1090/conm/412 chicago: Lieb, Élliott, Robert Seiringer, and Jan Solovej. “Ground-State Energy of a Dilute Fermi Gas,” 412:239–48. American Mathematical Society, 2006. https://doi.org/10.1090/conm/412. ieee: É. Lieb, R. Seiringer, and J. Solovej, “Ground-state energy of a dilute Fermi gas,” presented at the Differential Equations and Mathematical Physics, 2006, vol. 412, pp. 239–248. ista: Lieb É, Seiringer R, Solovej J. 2006. Ground-state energy of a dilute Fermi gas. Differential Equations and Mathematical Physics, Contemporary Mathematics, vol. 412, 239–248. mla: Lieb, Élliott, et al. Ground-State Energy of a Dilute Fermi Gas. Vol. 412, American Mathematical Society, 2006, pp. 239–48, doi:10.1090/conm/412. short: É. Lieb, R. Seiringer, J. Solovej, in:, American Mathematical Society, 2006, pp. 239–248. conference: name: Differential Equations and Mathematical Physics date_created: 2018-12-11T11:57:03Z date_published: 2006-01-01T00:00:00Z date_updated: 2021-01-12T06:56:51Z day: '01' doi: 10.1090/conm/412 extern: 1 intvolume: ' 412' main_file_link: - open_access: '1' url: http://arxiv.org/abs/math-ph/0507049 month: '01' oa: 1 page: 239 - 248 publication_status: published publisher: American Mathematical Society publist_id: '4593' quality_controlled: 0 status: public title: Ground-state energy of a dilute Fermi gas type: conference volume: 412 year: '2006' ... --- _id: '2334' author: - first_name: Robert full_name: Robert Seiringer id: 4AFD0470-F248-11E8-B48F-1D18A9856A87 last_name: Seiringer orcid: 0000-0002-6781-0521 - first_name: Élliott full_name: Lieb, Élliott H last_name: Lieb - first_name: Jakob full_name: Yngvason, Jakob last_name: Yngvason citation: ama: 'Seiringer R, Lieb É, Yngvason J. One-dimensional behavior of dilute, trapped Bose gases in traps. In: Zambrini J, ed. World Scientific Publishing; 2006. doi:10.1007/s00220-003-0993-3' apa: 'Seiringer, R., Lieb, É., & Yngvason, J. (2006). One-dimensional behavior of dilute, trapped Bose gases in traps. In J. Zambrini (Ed.). Presented at the ICMP: International Congress on Mathematical Physics, World Scientific Publishing. https://doi.org/10.1007/s00220-003-0993-3' chicago: Seiringer, Robert, Élliott Lieb, and Jakob Yngvason. “One-Dimensional Behavior of Dilute, Trapped Bose Gases in Traps.” edited by Jean Zambrini. World Scientific Publishing, 2006. https://doi.org/10.1007/s00220-003-0993-3. ieee: 'R. Seiringer, É. Lieb, and J. Yngvason, “One-dimensional behavior of dilute, trapped Bose gases in traps,” presented at the ICMP: International Congress on Mathematical Physics, 2006.' ista: 'Seiringer R, Lieb É, Yngvason J. 2006. One-dimensional behavior of dilute, trapped Bose gases in traps. ICMP: International Congress on Mathematical Physics.' mla: Seiringer, Robert, et al. One-Dimensional Behavior of Dilute, Trapped Bose Gases in Traps. Edited by Jean Zambrini, World Scientific Publishing, 2006, doi:10.1007/s00220-003-0993-3. short: R. Seiringer, É. Lieb, J. Yngvason, in:, J. Zambrini (Ed.), World Scientific Publishing, 2006. conference: name: 'ICMP: International Congress on Mathematical Physics' date_created: 2018-12-11T11:57:03Z date_published: 2006-03-07T00:00:00Z date_updated: 2021-01-12T06:56:51Z day: '07' doi: 10.1007/s00220-003-0993-3 editor: - first_name: Jean full_name: Zambrini, Jean-Claude last_name: Zambrini extern: 1 main_file_link: - open_access: '1' url: http://arxiv.org/abs/math-ph/0305025 month: '03' oa: 1 publication_status: published publisher: World Scientific Publishing publist_id: '4592' quality_controlled: 0 status: public title: One-dimensional behavior of dilute, trapped Bose gases in traps type: conference year: '2006' ...