# What is the probability that a random integral quadratic form in n variables has an integral zero?

## Author(s): Bhargava, Manjul; Cremona, John E.; Fisher, Tom A.; Jones, NG; Keating, JP

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Abstract: | We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$. |

Publication Date: | 2016 |

Electronic Publication Date: | 9-Sep-2015 |

Citation: | M. Bhargava, J. E. Cremona, T. Fisher, N. G. Jones, J. P. Keati ng: What is the probability that a random integral quadratic form in n variables has an integral zero ? Int. Math. Res. Not. IMRN, 12 (2016), 3828–3848. |

Pages: | 3828-3848 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | INTERNATIONAL MATHEMATICS RESEARCH NOTICES |

Version: | Author's manuscript |

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