scholarly journals The Continued Fraction Expansion of An Algebraic Power Series Satisfying A Quartic Equation

1995 ◽  
Vol 50 (2) ◽  
pp. 335-344 ◽  
Author(s):  
M.W. Buck ◽  
D.P. Robbins
Keyword(s):  
Power Series ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Quartic Equation ◽  
Algebraic Power Series

scholarly journals Continued fractions and diophantine equations in positive characteristic

2021 ◽  
Vol 109 (123) ◽  
pp. 143-151
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Ring Of Polynomials ◽  
Algebraic Power Series

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

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.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

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