Continued fractions for certain algebraic power series.

*(English)*Zbl 0591.10021This paper is concerned with the theory of continued fractions over \(\mathbb F_ p((x^{-1}))\) (the field of formal Laurent series in \(x^{-1}\) over the field with \(p\) elements) the polynomials in \(x\) playing the role of the integers. Improving and generalizing results of L. E. Baum and M. M. Sweet [Ann. Math. (2) 103, 593–610 (1976; Zbl 0312.10024)] about the unique root \(f\) in \(\mathbb F_ 2((x^{-1}))\) of the cubic equation \(f=-x/(xf^ 2+1),\) the authors give some examples of elements of \(\mathbb F_ p((x^{-1})),\) algebraic over \(\mathbb F_ p(x)\), for which the degrees of the partial quotients are bounded (in all the examples they are linear). For all examples, including the Baum-Sweet one, the continued fraction expansion is explicitly given. All calculations are based on a process that allows to obtain the continued fraction expansion of \((Rf^ p+S)/(Tf^ p+U)\) from that of \(f\) when \(R, S, T\) and \(U\) are polynomials.

Reviewer: Gilles Christol (Paris)

##### MSC:

11J70 | Continued fractions and generalizations |

11J61 | Approximation in non-Archimedean valuations |

11T55 | Arithmetic theory of polynomial rings over finite fields |

##### Keywords:

cubic equation; algebraic power series; finite fields; field of formal Laurent series; continued fraction expansion
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\textit{W. H. Mills} and \textit{D. P. Robbins}, J. Number Theory 23, 388--404 (1986; Zbl 0591.10021)

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##### References:

[1] | Baum, L.E; Sweet, M.M, Continued fractions of algebraic power series in characteristic 2, Ann. of math., 103, 593-610, (1976) · Zbl 0312.10024 |

[2] | Baum, L.E; Sweet, M.M, Badly approximable power series in characteristic 2, Ann. of math., 105, 573-580, (1977) · Zbl 0352.10017 |

[3] | Beynon, W.M, A formal account of some elementary continued fraction algorithms, J. algorithms, 4, 221-240, (1983) · Zbl 0522.68038 |

[4] | Raney, G.N, On continued fractions and finite automata, Math. ann., 206, 265-283, (1973) · Zbl 0251.10024 |

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