Continued Fractions and Transcendence of Formal Power Series Over a Finite Field

2014 ◽  
Vol 13 (2) ◽  
pp. 527-536
Author(s):  
B. Ammous ◽  
S. Driss ◽  
M. Hbaib
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fractions ◽  
Formal Power

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 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.

scholarly journals Irreducibility criterion and 2-pisot series in positive character

2019 ◽  
Vol 105 (119) ◽  
pp. 151-159
Author(s):  
Ammar Ben ◽  
Hassen Kthiri
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Unit Disc ◽  
Sufficient Condition ◽  
Irreducibility Criterion ◽  
Formal Power ◽  
Positive Character

Let Fq((X?1)) be the field of formal power series over a finite field Fq. We characterize a pair of roots that lies outside the unit disc while all remaining conjugates have a modulus strictly less than 1. In particular, we provide a sufficient condition for a pair of formal power series to be a 2-Pisot series. We also give an irreducibility criterion over Fq [X].

scholarly journals Engel Series Expansions of Laurent Series and Hausdorff Dimensions

2003 ◽  
Vol 75 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Laurent Series ◽  
Series Expansions ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Formal Power

AbstractFor any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

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