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


Published:20110831
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
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 $P(Y)=Y^{n}\alpha XY^{n1}\alpha^n,$ 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.
MSC Classifications: 
11A55, 11D45, 11D72, 11J61, 11J66 show english descriptions
Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] Counting solutions of Diophantine equations Equations in many variables [See also 11P55] Approximation in nonArchimedean valuations unknown classification 11J66
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 nonArchimedean valuations 11J66  unknown classification 11J66
