Rational lines on cubic hypersurfaces
Consider the cubic hypersurface defined by \(c_1z^3_1+\cdots+ c_sz^3_s= 0\), where \(c_1,\dots, c_s\) are nonzero integers. Let \((x_1,\dots,x_s)\), \((y_1,\dots,y_s)\in Z^s\). It is clear that the affine line \((x_1,\dots, x_s)+ t(y_1,\dots, y_s)\) is contained in the above hypersurface if and only if \[ \sum^s_{j=1} c_jx^3_j= \sum^s_{j=1} c_jx^2_j y_j= \sum^s_{j=1} c_jx_jy^2_j= \sum^s_{j=1} c_jy^3_j= 0. \] Let \(N_s(P)\) denote the number of integral solutions of this system of equations in the box \([-P,P]^s\). \textit{S. T. Parsell} [Trans. Am. Math. Soc. 352, No. 11, 5045--5062 (2000; Zbl 1108.11302)] showed that \(N_s(P)\gg P^{2s-12}\) for sufficiently large \(P\) as soon as \(s\geq 55\). Later the same author [Mathematika 58, No. 2, 209--224 (2012; Zbl 1284.11059)] proved that \(N_s(P)=\sigma P^{2s-12}+ O(P^{2s-12-\delta})\) for some positive \(\sigma\) and \(\delta\), depending on \(s\) and the \(c_j\)'s as soon as \(s\geq 29\). In the paper under review, the author establishes this asymptotic formula for \(s\geq 21\). The new ingredient of the proof is a mean-value estimate for \[ \sum_{1\leq x\leq P}\,\sum_{1\leq y\leq P} e(\alpha_1 x^3+ \alpha_2 x^2 y+ \alpha_3 xy^2+ \alpha_4 y^3) \] by considering cancellations from both the summation over \(x\) and the summation over \(y\).
scientific article; zbMATH DE number 6673188
scientific article