# Statistics of random integral matrices

## Abstract/Contents

- Abstract
- This thesis consists of two separate projects. The first part investigates the asymptotic behavior of the number of integral m-by-n matrices, with entries bounded by T, whose cokernel is isomorphic to a fixed abelian group G. We answer this question by building on work of Katznelson, who obtained asymptotics on the number of such matrices of given rank. In particular, we show that if G has torsion B for a finite abelian group B, then a positive proportion of matrices of rank r have cokernel isomorphic to G, and we compute this proportion explicitly (as an infinite product over primes). The corresponding problem for symmetric matrices is also discussed, with a different answer. Part two of this thesis deals with the infinitesimal frequency of a monic polynomial appearing as the characteristic polynomial of an n-by-n matrix with coefficients in the p-adic integers. Relying on the concept of rational singularities, we prove that this frequency is described by a continuous density on the space of monic polynomials, and show that the normalized density function is multiplicative, thus reducing its computation to the case of a monic irreducible polynomial. In the monic irreducible case, we express the density function as a finite sum over modules in the ring of integers of a finite extension of Q_p, and compute it in the case of degree < =3. For the general case, we conjecture bounds on the size of this function, as well as conjectures on the underlying geometric structures. In the end, we study a modification of this question as n goes to infinity. As an aside, we also use Cohen-Lenstra measures to compute the distribution of the Jordan blocks of matrices with coefficients in a fixed finite field as the size goes to infinity.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2016 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Boreico, Iurie | |
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Associated with | Stanford University, Department of Mathematics. | |

Primary advisor | Venkatesh, Akshay, 1981- | |

Thesis advisor | Venkatesh, Akshay, 1981- | |

Thesis advisor | Soundararajan, Kannan, 1973- | |

Thesis advisor | Yun, Zhiwei, 1982- | |

Advisor | Soundararajan, Kannan, 1973- | |

Advisor | Yun, Zhiwei, 1982- |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Iurie Boreico. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis (Ph.D.)--Stanford University, 2016. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2016 by Iurie Boreico

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