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

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 $O\left({\left(\frac{1}{2}(r+s-1{)}^{\delta }(r+s{)}^{1+\frac{1}{2(r+s-1)}}\right)}^{r+s-1}\right)\cdot {\left(e^{1+\frac{1}{2e}}\right)}^{r+s}(r+s)!$, where $\delta =1/2$ if $r+s\le 11$ or $\delta =1$ otherwise. The second result states that there exists a linear time algorithm to reduce a totally positive unary form $ax{x}^{\ast }$, 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.

15 pages, including abstract and bibliography