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The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 2 / 01 June 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 258-264
- Print publication:
- 01 June 2013
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- Article
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Dufresnoy and Pisot characterized the smallest Pisot number of degree
$n\,\ge \,3$ 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 
$\left( \text{SPE} \right)$ of degree 
$n$ in the field of formal power series over a finite field is given by 
$P\left( Y \right)\,=\,{{Y}^{n}}\,-\,\alpha X{{Y}^{n-1}}\,-{{\alpha }^{n}}$ where 
$\alpha $ is the least element of the finite field 
${{\mathbb{F}}_{q}}\backslash \left\{ 0 \right\}$ (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.
