Counting rational points on hypersurfaces

article Counting rational points on hypersurfaces published Timothy D Browning author 35827D50-F248-11E8-B48F-1D18A9856A870000-0002-8314-0177 Roger Heath Brown author For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε. Walter de Gruyter and Co 2005 eng Journal fur die Reine und Angewandte Mathematik 0075-4102 abs/math/0404456https://doi.org/10.1515/crll.2005.2005.584.83 58483 - 115 yes T.D. Browning, R. Heath Brown, Journal Fur Die Reine Und Angewandte Mathematik (2005) 83–115. Browning TD, Heath Brown R. Counting rational points on hypersurfaces. Journal fur die Reine und Angewandte Mathematik. 2005;(584):83-115. doi:<a href="https://doi.org/10.1515/crll.2005.2005.584.83">https://doi.org/10.1515/crll.2005.2005.584.83</a> Browning, Timothy D, and Roger Heath Brown. “Counting Rational Points on Hypersurfaces.” Journal Fur Die Reine Und Angewandte Mathematik. Walter de Gruyter and Co , 2005. <a href="https://doi.org/10.1515/crll.2005.2005.584.83">https://doi.org/10.1515/crll.2005.2005.584.83</a>. T. D. Browning and R. Heath Brown, “Counting rational points on hypersurfaces,” Journal fur die Reine und Angewandte Mathematik, no. 584. Walter de Gruyter and Co , pp. 83–115, 2005. Browning TD, Heath Brown R. 2005. Counting rational points on hypersurfaces. Journal fur die Reine und Angewandte Mathematik. (584), 83–115. Browning, T. D., & Heath Brown, R. (2005). Counting rational points on hypersurfaces. Journal Fur Die Reine Und Angewandte Mathematik. Walter de Gruyter and Co . <a href="https://doi.org/10.1515/crll.2005.2005.584.83">https://doi.org/10.1515/crll.2005.2005.584.83</a> Browning, Timothy D., and Roger Heath Brown. “Counting Rational Points on Hypersurfaces.” Journal Fur Die Reine Und Angewandte Mathematik, no. 584, Walter de Gruyter and Co , 2005, pp. 83–115, doi:<a href="https://doi.org/10.1515/crll.2005.2005.584.83">https://doi.org/10.1515/crll.2005.2005.584.83</a>. 2122018-12-11T11:45:14Z2026-08-21T09:31:08Z