Formal power series and their continued fraction expansion

1998 ◽  
pp. 358-371 ◽  
Author(s):  
Alf van der Poorten
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Formal Power

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

2013 ◽  
Vol 56 (2) ◽  
pp. 258-264
Author(s):  
A. Chandoul ◽  
M. Jellali ◽  
M. Mkaouar
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Laurent Series ◽  
Minimal Polynomial ◽  
Ordered Set ◽  
Pisot Number ◽  
Continued Fraction Expansion ◽  
Formal Power

Abstract.Dufresnoy and Pisot characterized the smallest Pisot number of degree n ≥ 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) = Yn–XYn-1–αn where α is the least element of the finite field 픽q\{0} (as a finite total ordered set). We prove that the sequence of SPEs of degree n is decreasing and converges to αX: Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

ON TRANSCENDENTAL CONTINUED FRACTIONS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS

2021 ◽  
pp. 1-12
Author(s):  
BÜŞRA CAN ◽  
GÜLCAN KEKEÇ
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Finite Fields ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansions ◽  
Formal Power

Abstract In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$ -numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$ .

scholarly journals Convergence of β-Modified Jacobi-Perron Algorithm over the Field of Formal Power Series

2019 ◽  
Vol 12 (2) ◽  
pp. 418-431
Author(s):  
Amara Chandoul ◽  
Fahad Aljuaydi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Formal Power ◽  
Fraction Algorithm

The aim of this paper is to study multidimentional $\beta$-continued fraction algorithm over the field of formal power series. In the case of the Modified Jacobi-Perron algorithm, we prove that it converges.

scholarly journals Transcendental continued β-fraction with quadratic pisot basis over Fq((x-1))

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4585-4591
Author(s):  
Marwa Gouadri ◽  
Mohamed Hbaib
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Formal Power

Let Fq be a finite field and Fq((x-1)) is the field of formal power series with coefficients in Fq. Let ??Fq((x-1)) be a quadratic Pisot series with deg(?) = 2. We establish a transcendence criterion depending on the continued ?-fraction of one element of Fq((x-1)).

scholarly journals A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

2021 ◽  
Vol 27 (1) ◽  
pp. 115-122
Author(s):  
Rima Ghorbel ◽  
Hassen Kthiri
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Formal Power

Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

scholarly journals On The Geometric Continued Fractions in Positive Characteristic

2019 ◽  
Vol 25 (2) ◽  
pp. 139-145
Author(s):  
Sana Driss ◽  
Hassen Kthiri
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Characteristic ◽  
Formal Power

In this paper we study another form in the field of formal power series over a finite field. If the continued fraction of a formal power seriesin $\mathbb{F}_q((X^{-1}))$ begins with sufficiently largegeometric blocks, then $f$ is transcendental.

scholarly journals A transcendence criterion for continued fraction expansions in positive characteristic

2015 ◽  
Vol 98 (112) ◽  
pp. 237-242
Author(s):  
Basma Ammous ◽  
Sana Driss ◽  
Mohamed Hbaib
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Characteristic ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Formal Power

We exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients.

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