arXiv subjects

Alison Beth Miller

Publications and source records attributed to Alison Beth Miller.

5 recordsLinked to original sources

arXiv2023

Seifert surfaces in the four-ball and composition of binary quadratic forms

We use composition of binary quadratic forms to systematically create pairs of Seifert surfaces that are non-isotopic in the four-ball. Our main topological result employs Gauss composition to classify the pairs of binary quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert surfaces of genus one. The main ingredient of the proof is number-theoretic and of independent interest. It establishes a new connection between the Bhargava cube and the geometric approach to Gauss composition via planes in the space of two-by-two matrices. In particular, we obtain a geometric recipe that given any two binary quadratic forms finds a Bhargava cube that gives rise to their composition.

math.GT↗

arXiv2022

Two-variable polynomials with dynamical Mahler measure zero

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd--Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.

math.NT↗

arXiv2022

Dynamical Mahler Measure: A survey and some recent results

We study the dynamical Mahler measure of multivariate polynomials and present dynamical analogues of various results from the classical Mahler measure as well as examples of formulas allowing the computation of the dynamical Mahler measure in certain cases. We discuss multivariate analogues of dynamical Kronecker's Lemma and present some improvements on the result for two variables due to Carter, Lalín, Manes, Miller, and Mocz.

math.NT↗

arXiv2020

On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime

In this paper we obtain an asymptotic formula for the number of $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\bZ$ having bounded discriminant $\Delta = 1-4p$, with $p$ a prime. We also give a random Euler product model for the distribution of Hurwitz class numbers, which is supported by our formula.

math.NT↗

arXiv2020

Asymptotics for the number of Simple $(4a+1)$-Knots of Genus 1

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of $\mathbb{Z}[1/m]$-equivalence classes of binary quadratic forms of discriminant $1-4m$, for $m$ running through the same range. Our heuristics, based on the Cohen-Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$, and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ prime is bounded above by $O(X^{3/2}/\log X)$, and that the total itself is $o(X^{3/2})$.

math.NT↗