Least zero of pairs of additive cubic equations

Abstract

An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 = 0, \end{cases} \] under the "$M$-good" condition for $n \ge 16$, where $c_{1}, \dots, c_{n}$ and $d_{1}, \dots, d_{n}$ are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the $M$-good condition.