Pisot Numbers in the Neighborhood of a Limit Point. II

Abstract

Let S denote the set of real algebraic integers greater than one, all of whose other conjugates lie within the unit circle. In an earlier paper, we introduced the notion of "width" of a limit point $\alpha$ of S and showed that, if the width of $\alpha$ is smaller than 1.28... then there is an algorithm for determining all members of S in a neighborhood of $\alpha$. Recently, we introduced the "derived tree" in order to deal with limit points of greater width. Here, we apply these ideas to the study of the limit point ${\alpha _3}$, the zero of ${z^4} - 2{z^3} + z - 1$ outside the unit circle. We determine the smallest neighborhood ${\theta _1} < {\alpha _3} < {\theta _2}$ of ${\alpha _3}$ in which all elements of S other than ${\alpha _3}$ satisfy one of the equations ${z^n}({z^4} - 2{z^3} + z - 1) \pm A(z) = 0$, where $A(z)$ is one of ${z^3} - {z^2} + 1$, ${z^3} - z + 1$ or ${z^4} - {z^3} + z - 1$. The endpoints ${\theta _1}$, and ${\theta _2}$ are elements of S of degrees 23 and 42, respectively.