Cubic forms as sum of cubes of linear forms (Q1277200)

Consider the set \(V_n\) of cubic forms \(F(x)= \sum_{1\leq i,j,k\leq n}\alpha_{ijk} x_ix_jx_k\) of \(n\) variables with integral rational coefficients \(\alpha_{ijk}\), representable as a sum of cubes of two linear forms \(L_1(x)\), \(L_2(x)\) with algebraic coefficients: \(F(x)= L_1(x)^3+ L_2(x)^3\), and let \(N(n,t)\) be the number of forms \(F\) in \(V_n\) satisfying the condition: \(| \alpha_{ijk}|\leq t\) for \(1\leq i,j,k\leq n\). Clearly, \(N(n,t)\gg t^{2n/3}\). The goal of this paper is an upper estimate. The author proves that if \(n\geq 10\) then \[ N(n,t)\ll t^{2n/3}, \] with the implied constant depending only on \(n\). Given a form \(F\) in \(V_n\), there is a unique square-free integer \(d\) such that \[ F(x)= \lambda \Biggl( \sum_{i=1}^n \alpha_ix_i \Biggr)^3+ \lambda' \Biggl( \sum_{i=1}^n \alpha_i'x_i \Biggr)^3 \] for some \(\lambda\), \(\lambda'\), \(\alpha_i\), \(\alpha_i'\) in \(\mathbb{Q}(\sqrt{d})\), where \(b'\) stands for the conjugate of \(b\) in \(\mathbb{Q} (\sqrt{d})\) when \(d\neq 1\). Therefore \(N(n,t)= \mathop {{\sum'} _d} N(n,d,t)\), the sum \(\mathop {{\sum'} _d}\) being extended over the square-free integers \(d\); here \(N(n,d,t)\) denotes the number of forms in \(V_n\) corresponding (in the above sense) to \(d\), whose coefficients lie in the cube \(| a_{ijk}|\leq t\). The author proves that \[ N(n,d,t)\ll t^{2n/3} h(d) C^{\omega(d)} | d|^{-n/6} \] for some fixed \(C>0\) if \(n\geq 7\), where \(h(d)\) and \(\omega(d)\) denote the class number of \(\mathbb{Q} (\sqrt{d})\) and the number of distinct prime factors of \(d\), respectively. The proof of this theorem is rather involved, and depends, in particular, on a lemma in the geometry of numbers proved by \textit{W. M. Schmidt} [Acta Arith. 70, 343-375 (1995; Zbl 0784.11055)]. Since \(h(d) C^{\omega(d)}\ll| d|^{1/2+ \varepsilon}\) for any \(\varepsilon>0\), the upper estimate for \(N(n,t)\) is a consequence of this theorem.