Representation of integers by a family of cubic forms in seven variables II

Abstract

In an earlier paper [4], we derived asymptotic formulas for the number of representations of zero and of large positive integers by the cubic forms in seven variables which can be written as $L_1(x_1,x_2,x_3) Q_1(x_1,x_2,x_3)+ L_2(x_4,x_5,x_6) Q_2(x_4,x_5,x_6) + a_7 x_7^3$ where $L_1$ and $L_2$ are linear forms, $Q_1$ and $Q_2$ are quadratic forms and $a_7$ is a non-zero integer and for which certain quantities related to $L_1Q_1$ and $L_2Q_2$ were non-zero. In this paper, we consider the case when one or both of these quantities is zero but $L_1Q_1$ and $L_2Q_2$ are still nondegenerate cubic forms in three variables.