## Abstract

We obtain an essentially optimal estimate for the moment of order

*of the exponential sum having argument αx***32/3**^{3}+βx^{2}. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds**10.**Original language | English |
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Pages (from-to) | 325-356 |

Number of pages | 32 |

Journal | Proceedings of the London Mathematical Society |

Volume | 110 |

Issue number | 2 |

Early online date | 21 Nov 2014 |

DOIs | |

Publication status | Published - Feb 2015 |

## Keywords

- math.NT
- 11D72
- 11L15
- 11L07
- 11P55
- Weyl sums
- Hardy-Littlewood method
- Diophantine equations