The density of rational points on non-singular hypersurfaces
type: Item
instance of: scholarly article
title: The density of rational points on non-singular hypersurfaces
zbMATH Open document ID: 0808.11042
DOI: 10.1007/BF02830871
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1995-03-09T00:00:00Z
review text: Let \(F({\mathbf x})= F[x_ 1,x_ 2,\ldots, x_ n]\in \mathbb{Z} [x_ 1, x_ 2,\ldots, x_ n]\) be a non-singular form of degree \(d\geq 2\), and let \[ N(F,X)= \#\{{\mathbf x}\in \mathbb{Z}^ n:\;F({\mathbf x})=0,\;|{\mathbf x}| \leq X\}, \] where \(|{\mathbf x}|= \max_{1\leq r\leq n} | x_ i|\). \textit{M. Fujiwara} [Number theory and combinatorics, Japan 1984, 89-96 (1985; Zbl 0606.10015)] proved that \(N(F,X)\ll X^{n- 2+2/n}\) for any fixed form \(F\). The author reduces the exponent to \(n- 2+2/ (n+1)\) for \(n\geq 4\) and to \(n-3+15/ (n+5)\) for \(n\geq 8\) and \(d\geq 3\). It is conjectured that \(n-2+ \varepsilon\) is admissible for \(n\geq 3\), and the latter result establishes this conjecture for \(n\geq 10\). The proof uses Deligne's bounds for exponential sums and for the number of points on hypersurfaces over finite fields. The author uses the trivial inequality \(N(F,X)\leq N(F,X,m)\), where \[ N(F,X,m)= \#\{{\mathbf x}\in \mathbb{Z}^ n:\;F({\mathbf x})\equiv 0\pmod m,\;|{\mathbf x}|\leq X\}. \] The exponent \(n-2+ 2/(n+1)\) is obtained using a prime modulus. The exponent \(n-3+15/ (n+5)\) is obtained with a composite modulus and a ``\(q\)-analog'' of van der Corput's method.
Mathematics Subject Classification ID: 11G99
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 562800
zbMATH Keywords: rational points
zbMATH Keywords: hypersurface
zbMATH Keywords: counting function
zbMATH Keywords: multiple exponential sum
zbMATH Keywords: singular locus
zbMATH Keywords: Deligne's bounds for exponential sums
zbMATH Keywords: number of points
zbMATH Keywords: hypersurfaces over finite fields
author: D. R. Heath-Brown
reviewed by: Sidney W. Graham
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 562800