# Representation of integers by a family of cubic forms II

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## Abstract

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 Classification

Primary 11D45 Secondary 11D85 11P55## Notes

### Acknowledgments

I would like to thank Prof. Robert C. Vaughan for many useful discussions and the referee for useful comments and suggestions.

## References

- 1.Franke, J., Manin, YuI, Tschinkel, Yu.: Rational points of bounded height on Fano varieties. Invent. Math.
**95**(2), 421–435 (1989)MathSciNetCrossRefGoogle Scholar - 2.Vaughan, R.C.: The Hardy–Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997)Google Scholar
- 3.Vaughan, R.C.: Integer points on elliptic curves. Rocky Mt. J. Math. (to appear).Google Scholar
- 4.Verma, M.: Representation of integers by a family of cubic forms. Ramanujan J.
**34**(1), 11–38 (2014)Google Scholar