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

TL;DR

This paper investigates the structure of Galois number fields, providing bounds on the fundamental domain's facets, an efficient reduction algorithm for totally positive forms, and bounds on the Weil height of Pisot units, with implications for cryptography.

Contribution

It introduces bounds on the fundamental domain facets, a linear-time reduction algorithm for positive forms, and height bounds for Pisot units in Galois extensions, advancing number theory and cryptography.

Findings

Bound on the number of facets of the fundamental domain.

Linear time algorithm for reducing totally positive unary forms.

Upper bound on the Weil height of the shortest Pisot unit.

Abstract

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 $O\left(\left(\frac{1}{2}(r+s-1)^\delta(r+s)^{1+\frac{1}{2(r+s-1)}}\right)^{r+s-1}\right) \cdot\left(e^{1+\frac1{2e}}\right)^{r+s}(r+s)!$, where $\delta=1/2$ if $r+s \leq 11$ or $\delta=1$ 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 $\frac{1}{[K:\mathbb{Q}]}\left(\frac{\gamma}{2}(r+s-1)^{\delta-\frac{1}{2(r+s-1)}}R_K^{\frac{1}{r+s-1}}+(r+s-1)\epsilon\right)$, where $R_K$ denotes the regulator of $K$, $\gamma=1$ if $K$ is totally real or $2$ otherwise, and $\epsilon>0$ is some arbitrarily small constant.