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 On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

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 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.

scholarly journals A Study of Cyclic Codes Via a Surjective Mapping

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 325-332
Author(s):  
Mriganka Sekhar Dutta ◽  
Helen K. Saikia
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Surjective Mapping ◽  
Formal Power

In this article, cyclic codes of length $n$ over a formal power series ring and cyclic codes of length $nl$ over a finite field are studied. By defining a module isomorphism between $R^n$ and $(Z_4)^{2^kn}$, Dinh and Lopez-Permouth proved that a cyclic shift in $(Z_4)^{2^kn}$ corresponds to a constacyclic shift in $R^n$ by $u$, where $R=\frac{Z_4[u]}{<u^{2^k}-1>}$. We have defined a bijective mapping $\Phi_l$ on $R_{\infty}$, where $R_{\infty}$ is the formal power series ring over a finite field $\mathbb{F}$. We have proved that a cyclic shift in $(\mathbb{F})^{ln}$ corresponds to a $\Phi_l-$cyclic shift in $(R_{\infty})^n$ by defining a mapping from $(R_{\infty})^n$ onto $(\mathbb{F})^{ln}$. We have also derived some related results.

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)).

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic

2013 ◽  
Vol 56 (4) ◽  
pp. 673-683
Author(s):  
K. Ayadi ◽  
M. Hbaib ◽  
F. Mahjoub
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Diophantine Approximation ◽  
Positive Characteristic ◽  
Rational Approximations ◽  
Positive Degree ◽  
Formal Power ◽  
Algebraic Power Series

Abstract.In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the typewhere (A, B, C) ∊ (𝔽q[X])2 × 𝔽*q [X]. In particular, under some conditions on the polynomials A, B and C, we will give well approximated elements satisfying this equation.

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.

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