## Abstract

Let

*C*be a cubic form with integer coefficients in*n*variables, and let*h*be the*h*-invariant of*C*. Let*L*_{1},…,*L*be linear forms with real coefficients such that, if α∈ℝ_{r}*∖{0}, then α⋅*^{r}*L*is not a rational form. Assume that*h*>16+8*r*. Let τ∈ℝ*, and let*^{r}*η*be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions*x*∈[−*P*,*P*]*to the system*^{n}*C*(*x*)=0, |*L*(*x*)−τ|<*η*. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the*h*-invariant condition with the hypothesis*n*>16+9*r*and show that the system has an integer solution. Finally, we show that the values of LL at integer zeros of*C*are equidistributed modulo**1**in ℝ^{r}, requiring only that*h*>16.Original language | English |
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Pages (from-to) | 421-450 |

Number of pages | 30 |

Journal | Algebra and Number Theory |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 16 Mar 2016 |

## Keywords

- diophantine equations
- diophantine inequalities
- diophantine approximation
- equidistribution