Florin Stan: Siegel-type limit points above the PSZ curve, 95-103

Abstract:

Let $\mathcal{F}$ be the set of minimal polynomials of totally real and positive algebraic integers. An element $f$ of $\mathcal{F}$ can be written as $f=x^n-a_{n-1}x^{n-1}+\cdots+(-1)^n a_0$, with $a_i \in \Z^{+}$, for all $0 \le i \le n-1$. A Siegel-type point is a limit point of the set

$$\mathcal{P} = \Bigg\{ \left( \frac{d}{n}, \left( \frac{a_{n-d... .../d} \right) \in \R^2: f \in \mathcal{F}, 1 \le d \le n \Bigg\}$$

In this paper we investigate the location of these Siegel-type points and study some of their properties. We show that the points $(x,y)$ high enough in the strip $0<x<1$ are close to Siegel-type points.

Key Words: Totally positive algebraic integer, Schur-Siegel-Smyth trace problem.

2010 Mathematics Subject Classification: Primary 11R04; Secondary 11R06.