The Ramanujan Journal

, Volume 37, Issue 3, pp 597–615

# Representation of integers by a family of cubic forms II

• Manoj Verma
Article

## 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.

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