On a certain nonary cubic form and related equations
Let \(S_3(P; 1, 3)\) denote the number of solutions of the system \[ x^3+ y^3+ z^3= X^3+ Y^3+ Z^3,\quad x+ y+ z= X+ Y+ Z, \] \[ 1\leq x, y, z, X, Y, Z\leq P. \] \textit{L.-K. Hua} [Additive theory of prime numbers, Am. Math. Soc. (1965; Zbl 0192.39304)] showed \(S_3(P; 1,3)\ll P^3\log^9 P\), this being of importance in connection with Weyl's inequality an d Hua's lemma. By elementary but intricate methods the authors establish a much more precise result \(S_3(P; 1, 3)= 6P^3+ U(P)\), where \(P^2\log^5 P\ll U(P)\ll P^2\log^5 P\). Here, the term \(6P^3\) arises from the ``trivial'' solutions in which \(x\), \(y\), \(z\) are a permutation of \(X\), \(Y\), \(Z\). They suggest that their methods might lead to an asymptotic formula for the number \(U(P)\) of non-trivial solutions, but this has not yet been done, because of the further complications that would appear likely to be involved. Further, let \(S_s(P; k)\) denote the number of solutions of the system \(\sum^s_{i= 1} (x^{k_j}_i- y^{k_j}_i)= 0\) \((j= 1,\dots, t)\) with \(1\leq x_i\), \(y_i\leq P\). They record relatively weaker upper bounds (of the form \(\ll P^{s- \delta}\) with explicit \(\delta= \delta_s> 0\)) for the quantities analogous to \(U(P)\) in the two cases \(k= \{1, 2,\dots, s- 2, s\}\), \(k= \{1, 2,\dots, s- 1\}\), the last of which is a case of Vinogradov's mean value theorem.
scientific article; zbMATH DE number 871841
scientific article