Representation of integers by a family of cubic forms II
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In an earlier paper Verma (Ramanujan J 2014), 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 non-degenerate cubic forms in three variables.
Mathematics Subject ClassificationPrimary 11D45 Secondary 11D85 11P55
I would like to thank Prof. Robert C. Vaughan for many useful discussions and the referee for useful comments and suggestions.
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