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 $L_1{Q}_1$ and $L_2{Q}_2$ were non-zero. In this paper, we consider the case when one or both of these quantities is zero but $L_1{Q}_1$ and $L_2{Q}_2$ are still nondegenerate cubic forms in three variables.

arXiv comment13 pages