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Search: MSC category 11D72 ( Equations in many variables [See also 11P55] )

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1. CMB Online first

Nguyen, Khoa Dang
The Hermite-Joubert Problem and a Conjecture of Brassil-Reichstein
show that Hermite's theorem fails for every integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$ with integers $k_1\gt k_2\gt k_3\geq 0$. This confirms a conjecture of Brassil and Reichstein. We also obtain new results for the relative Hermite-Joubert problem over a finitely generated field of characteristic $0$.

Keywords:Hermite-Joubert problem, Brassil-Reichstein conjecture, diophantine equation
Categories:11D72, 11G05

2. CMB 2011 (vol 56 pp. 258)

Chandoul, A.; Jellali, M.; Mkaouar, M.
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
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
Categories:11A55, 11D45, 11D72, 11J61, 11J66

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