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

Abstract

2. Notations À cause des limitations du traitement de texte utilisé, les lettres N, Z, Q, R et C désigneront les ensembles usuels. Ces lettres pourront parfois avoir une autre signification, sans qu'une confusion soit possible. De même, nous utiliserons la lettre F pour désigner une partie finie de Z. La lettre P comme ensemble des indices d'une somme ou d'un produit désignera l'ensemble des nombres premiers. La notation [a, b], où a et b sont des entiers désignera l'ensemble {a, a + 1, ..., b}.

Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

“…for all PÂQ, implies that it is not of class I, according to the theorem proved in [5] or [7]. In the same paper [2], the authors have considered the unique solution, in K((T &1 )), of the algebraic equation (1), when the base field is K=F 13 . In that situation the solution is actually of class I.…”

“…In [2], Buck and Robbins have given the continued fraction expansion of an element of F 3 ((T &1 )). If K=F 3 , they show that the algebraic equation…”

“…For k>0, we put d k = deg a k and P k ÂQ k =[a 0 , ..., a k ]. It results, from the inductive definition (2), that all partial quotients are monomials, and all have a power of 3 as the degree. For i 1, we define…”

“…This new approach allows us to give another proof of the result due to Buck and Robbins [2]. Let : be an irrational element of K((T &1 )).…”

“…Buck and Robbins [2] have given the continued fraction expansion of a particular algebraic element of F 3 ((T &1 )). What is very curious in this example is that it does not belong to the set of exceptions already known.…”