The density of rational points on non-singular hypersurfaces
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.
scientific article; zbMATH DE number 562800
scientific article