arXiv subjects

Alar Leibak

Publications and source records attributed to Alar Leibak.

3 recordsLinked to original sources

arXiv2023

Unit Reducible Cyclotomic Fields

In this paper, we continue the study of unit reducible fields as introduced in \cite{LPL23} for the special case of cyclotomic fields. Specifically, we deduce that the cyclotomic fields of conductors $2,3,5,7,8,9,12,15$ are all unit reducible, and show that any cyclotomic field of conductor $N$ is not unit reducible if $2^4, 3^3, 5^2, 7^2, 11^2$ or any prime $p \geq 13$ divide $N$, meaning the unit reducible cyclotomic fields are finite in number. Finally, if $a$ is a totally positive element of a cyclotomic field, we show that for all equivalent $a^\prime$, the discrepancy between $\trace_{K/\mathbb{Q}}(a^\prime)$ and the shortest nonzero element of the quadratic form $\trace_{K/\mathbb{Q}}(axx^*)$ where $x$ is taken from the ring of integers tends to infinity as the conductor $N$ goes to infinity.

math.NT↗

arXiv2023

On Pisot Units and the Fundamental Domain of Galois Extensions of $\mathbb{Q}$

In this paper, we present two main results. Let $K$ be a number field that is Galois over $\mathbb{Q}$ with degree $r+2s$, where $r$ is the number of real embeddings and $s$ is the number of pairs of complex embeddings. The first result states that the number of facets of the reduction domain (and therefore the fundamental domain) of $K$ is no greater than $O\left(\left(\frac{1}{2}(r+s-1)^δ(r+s)^{1+\frac{1}{2(r+s-1)}}\right)^{r+s-1}\right) \cdot\left(e^{1+\frac1{2e}}\right)^{r+s}(r+s)!$, where $δ=1/2$ if $r+s \leq 11$ or $δ=1$ otherwise. The second result states that there exists a linear time algorithm to reduce a totally positive unary form $axx^*$, such that the new totally positive element $a^\prime$ that is equivalent to $a$ has trace no greater than a constant multiplied by the integer minimum of the trace-form $\trace(axx^*)$, where the constant is determined by the shortest Pisot unit in the number field. This may have applications in ring-based cryptography. Finally, we show that the Weil height of the shortest Pisot unit in the number field can be no greater than $\frac{1}{[K:\mathbb{Q}]}\left(\fracγ{2}(r+s-1)^{δ-\frac{1}{2(r+s-1)}}R_K^{\frac{1}{r+s-1}}+(r+s-1)ε\right)$, where $R_K$ denotes the regulator of $K$, $γ=1$ if $K$ is totally real or $2$ otherwise, and $ε>0$ is some arbitrarily small constant.

math.NT↗

arXiv2022

Unit Reducible Fields and Perfect Unary Forms

In this paper, we introduce the notion of unit reducibility for number fields, that is, number fields in which all positive unary forms attain their nonzero minimum at a unit. Furthermore, we investigate the link between unit reducibility and the number of homothety classes of perfect unary forms for a given number field, and prove an open conjecture about the number of classes of perfect unary forms in real quadratic fields, stated by D. Yasaki.

math.NT↗