arXiv subjects

Ester Sgallová

Publications and source records attributed to Ester Sgallová.

4 recordsLinked to original sources

arXiv2024

Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables

We study a new connection between multidimensional continued fractions, such as Jacobi--Perron algorithm, and additively indecomposable integers in totally real cubic number fields. First, we find the indecomposables of all signatures in Ennola's family of cubic fields, and use them to determine the Pythagoras numbers. Second, we compute a number of periodic JPA expansions, also in Shanks' family of simplest cubic fields. Finally, we compare these expansions with indecomposables to formulate our conclusions.

math.NT↗

arXiv2023

The Six-Vertex Model with a Non-Standard Boundary Condition

We consider the enumeration of states in the Brubaker-Bump-Friedberg six-vertex model, whose boundary conditions are determined by an integer partition. In general, we find the number of states is a polynomial in the largest part of the partition. By explicating this technique, we also enumerate the states completely for hook shapes and staircases.

math.CO↗

arXiv2022

Pythagoras numbers of orders in biquadratic fields

We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt5, \sqrt{s})$ we prove $\mathcal{P}(\mathcal{O}_K)=5$. Further we show that $5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.

math.NT↗

arXiv2022

Controlling distribution of prime sequences in discretely ordered principal ideal subrings of $\mathbb Q[x]$

We show how to construct discretely ordered principal ideal subrings of $\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ of positive integers such that, for each prime integer $p$, the set $\{p\mathbb Z, d_1+p\mathbb Z,\dots, d_l+p\mathbb Z\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\dots, d_l)\in \mathcal D$, a cofinal set of progressions $(f, f+d_1, \dots, f+d_l)$ of prime elements in our principal ideal domain $R_τ$. Moreover, we can simultaneously guarantee that each positive prime $g\in R_τ\setminus\mathbb N$ is either in a prescribed progression as above or there is no other prime $h$ in $R_τ$ such that $g-h\in\mathbb Z$. Finally, all the principal ideal domains we thus construct are non-Euclidean and isomorphic to subrings of the ring $\hat{\mathbb{Z}}$ of profinite integers.

math.NT↗