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# The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field

Published:2011-08-31
Printed: Jun 2013
• A. Chandoul,
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisia
• M. Jellali,
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisia
• M. Mkaouar,
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisia
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## Abstract

Dufresnoy and Pisot characterized the smallest Pisot number of degree $n \geq 3$ by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot's result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree $n$ in the field of formal power series over a finite field is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$ is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$ (as a finite total ordered set). We prove that the sequence of SPEs of degree $n$ is decreasing and converges to $\alpha X.$ Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.
 Keywords: Pisot element, continued fraction, Laurent series, finite fields
 MSC Classifications: 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11D45 - Counting solutions of Diophantine equations 11D72 - Equations in many variables [See also 11P55] 11J61 - Approximation in non-Archimedean valuations 11J66 - unknown classification 11J66