Number Theory

In 2014 I completed a Ph.D. in Mathematics at the Universitat Autònoma de Barcelona. My research is based on the OM representations of prime ideals produced by the Montes Algorithm, which I worked with to more efficiently compute bases of integral closures. My thesis advisors were Enric Nart and Jesús Montes.

You can download a copy of my thesis disseration and watch an animation of it being written on the Thesis page. The preprint Triangular bases of integral closures contains the main points of the thesis is a more concise format.


Triangular bases of integral closures

Hayden D. Stainsby

Abstract: In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings.

Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and let $\mathfrak{m}$ be the maximal ideal. We take $f \in \mathcal{O}[x]$, a monic irreducible polynomial of degree $n$ and consider the extension $L = K[x]/(f(x))$ as well as $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}$ in $L$, which we suppose to be finitely generated as an $\mathcal{O}$-module.

The algorithm $\operatorname{MaxMin}$, presented in this paper, computes triangular bases of fractional ideals of $\mathcal{O}_{L}$. The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions.

Published Articles

Complexity of OM Factorizations of Polynomials Over Local Fields

Jens-Dietrich Bauch, Enric Nart, Hayden D. Stainsby

Abstract: Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $\mathcal{O}$ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in\mathcal{O}[x]$ a monic separable polynomial of degree $n$. Let $\delta=v(\operatorname{Disc}(F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F \pmod{\mathfrak{m}^{\nu}}$, for a prescribed precision $\nu$. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O(n^{2+\epsilon}+n^{1+\epsilon}\delta^{2+\epsilon}+n^2\nu^{1+\epsilon})$ word operations for the complexity of the computation of a factorization of $F \pmod{\mathfrak{m}^{\nu}}$, assuming that the residue field of $k$ is small.