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.

Some remarks on formal power series and formal Laurent series

2017 ◽  
Vol 67 (3) ◽  
Author(s):  
Dariusz Bugajewski ◽  
Xiao-Xiong Gan
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Formal Power

AbstractIn this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.:

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.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

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].

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 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)).

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