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

Canadian mathematical bulletin

M. Mkaouar¹  M. Jellali¹  A. Chandoul¹ 

[1] Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisia

关键词: Pisot element;    continued fraction;    Laurent series;    finite fields;   

DOI  :  10.4153/CMB-2011-168-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Dufresnoy and Pisot characterized the smallestPisot number of degree $n geq 3$ by giving explicitly its minimalpolynomial. In this paper, we translate Dufresnoy and Pisot'sresult to the Laurent series case. Theaim of this paper is to prove that the minimal polynomialof the smallest Pisot element (SPE) of degree $n$ in the field offormal power series over a finite fieldis given by $P(Y)=Y^{n}-alpha XY^{n-1}-alpha^n,$ where $alpha$is the least element of the finite field $mathbb{F}_{q}ackslash{0}$(as a finite total ordered set). We prove that the sequence ofSPEs of degree $n$ is decreasing and converges to $alpha X.$Finally, we show how to obtain explicit continued fractionexpansion of the smallest Pisot element over a finite field.

【 授权许可】

Unknown