Monogenic Pisot and Anti-Pisot Polynomials

Abstract

A Pisot number is a real algebraic integer $\alpha \gt 1$ such that all of its Galois conjugates, other than $\alpha$ itself, lie inside the unit circle. An anti-Pisot number is a real algebraic integer $\alpha \gt 1$, such that exactly one Galois conjugate of $\alpha$ lies inside the unit circle, and $\alpha$ has at least one Galois conjugate, other than $\alpha$ itself, outside the unit circle. We call the minimal polynomials of these algebraic integers, respectively, Pisot and anti-Pisot polynomials. In this article, we find infinite families of Pisot (anti-Pisot) polynomials $f(x)$ such that $\{ 1, \alpha, \alpha^2, \ldots, \alpha^{n-1} \}$ is a basis for the ring of integers of $\mathbb{Q}(\alpha)$, where $\alpha$ is a Pisot (anti-Pisot) number and $\deg(f) = n$, for certain $n$. We refer to these polynomials as monogenic Pisot (anti-Pisot) polynomials.