On The Geometric Continued Fractions in Positive Characteristic

Sana Driss,Hassen Kthiri

doi:10.22342/jims.25.2.525.139-145

Abstract

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.

Highlights

The diophantine approximation issue introduced by Khintchine in [8] checks whether an irrational algebraic number x of degree > 2 has a continued fraction expansion with unbounded partial quotients
Xα3 + α + X = 0 has a continued fraction expansion with partial quotients of bounded degree. They observed that no real algebraic number of degree ≥ 3 has yet been shown to have bounded or unbounded partial quotients
The present paper is organized as follows: In section 2, we introduce the field of formal power series and the continued fraction expansion over this field and we review some basic properties

Summary

Introduction

The diophantine approximation issue introduced by Khintchine in [8] checks whether an irrational algebraic number x of degree > 2 has a continued fraction expansion with unbounded partial quotients. 1. INTRODUCTION The diophantine approximation issue introduced by Khintchine in [8] checks whether an irrational algebraic number x of degree > 2 has a continued fraction expansion with unbounded partial quotients. Maillet [10], was the first to explore transcendental continued fractions with unbounded partial quotients. It is clear that the methods used in [1, 2] do not enable one to study a formal power series whose continued fraction expansion satisfies a specific property.

Results

Conclusion