This paper considers lines \(L\) lying in a given diagonal cubic hypersurface defined over the rationals. Since any such hypersurface with \(s\geq 7\) has a rational point, by the work of R. C. Baker [Acta Arith. 53, No. 3, 217–250 (1989; Zbl 0642.10041)], it is easy to see that there will be a rational line as long as \(s\geq 14\). The present paper is concerned with estimating how frequent such lines are. Let \(N_s(P)\) denote the number of lines \(L\) lying in the variety, and taking the form \(t{\mathbf a}+{\mathbf b}\) with integer vectors \({\mathbf a}\), \({\mathbf b}\) satisfying \(\max(|{\mathbf a}|_\infty, |{\mathbf b}|_\infty)\leq P\). Lines which have more that one admissible representation (a possibility the author appears to have overlooked) are counted with appropriate multiplicity. The result of the paper is then that \(N_s(P)\gg P^{2s-12}\) if \(s\geq 57\). Here \(P^{2s-12}\) is the conjectural order of magnitude for \(N_s(P)\).
The proof applies the circle method to estimate the number of simultaneous solutions of the equations \[ \sum^s_{i=1} c_i x^3_i= \sum^s_{i=1} c_i x^2_i y_i= \sum^s_{i=1} c_i x_i y^2_i= \sum^s_{i=1} c_i y^3_i= 0. \] This uses various mean value estimates, the starting point for which is a paucity result for the simultaneous equations \[ \sum^3_{i=} x^3_i= \sum^6_{i=4} x^3_i,\quad \sum^3_{i=} x^2_i y_i= \sum^6_{i=4} x^2_i y_i,\quad\sum^3_{i=} x_i y^2_i= \sum^6_{i=4} x_i y^2_i,\quad \sum^3_{i=} y^3_i= \sum^6_{i=4} y^3_i. \]