Continued fraction and Diophantine equation
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.
Abstract
Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.
참고문헌 (19)
- Y. Amice, "Les nombres p-adiques", PUF Collection Sup
- P. Bateman, "The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series", Illinois J. Math, 6, pp.594~606 (1962)
- L. E. Baum, "Continued fraction of algebraic power series in charac-teristic 2", Ann. of Math, 103(3), pp.593~610 (1976)
- M. J. Bertin, "Pisot and Salem Numbers" (1992), Birkhauser Verlag Basel
- H. Davenport, "A remark on continued fractions", Michigan Math. J, 11, pp.343~344 (1964)
- M. Grandet-Hugot, "Sur une propriete des nombres de Pisot dans un corps de s´erie formelles", C.R.A.S. Ser. A, 266, pp.A39~A41 (1967)
- M. Grandet-Hugot, "Elements algebriques remarquables dans un corps de serie formelles", Acta Arith, 14, pp.177~184 (1968)
- G. H. Hardy, "An Introduction to the Theory of Numbers" (1979), Oxford Science Publication
- A. Lasjaunias, "Diophantine approximation and continued fraction expansions of alge-braic power series in positive characteristic", J. Number Theory, 65(2), pp.206~225 (1997)
- K. Mahler, "On a theorem of Liouville in fields of positive characteristic", Can. J. Math, 1, pp.397~400 (1949)
- B. de Mathan, "Approximation exponents for algebraic functions", Acta Arith, 60(4), pp.359~370 (1992)
- W. Mills, "Continued fraction for certain algebraic power series", J. Number Theory, 23(3), pp.388~404 (1986)
- M. Mkaouar, "Sur les fractions continues des s´eries formelles quadratiques sur Fq(X)", Acta Arith, 97(3), pp.241~251 (2001)
- R. P. Pass, "On the partial quotients of algebraic Integers", J. Number Theory, 11(1), pp.14~15 (1979)
- K. F. Roth, "Rational approximations to algebraic numbers", Mathematika, 2, pp.1~20 (1955)
- V. G Sprindzuk, "Mahler’s problem in metric number theory", Translation of Mathemat-ical monographs, 25 (1969), Amer. Math. Soc.
- D. Thakur, "Diophantine approximation exponents and continued fractions for algebraic power series", J. Number Theory, 79(2), pp.284~291 (1999)
- D. Thakur, "Diophantine approximation and transcendance in finite characteristic, (Pub-lished in the book Diophantine equations)"
- J. F. Voloch, "Diophantine approximation in positive characteristic", Period. Math. Hungar, 19(3), pp.217~225 (1988)