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Continued Fractions for Certain Algebraic Power Series over a Finite Field

  • Alain Lasjaunias

Abstract

In this survey we discuss rational approximation properties of certain algebraic power series over a finite field using continued fractions. These algebraic elements are fixed points of the composition of a linear fractional transformation and of the Frobenius homomorphism.

Keywords

Power Series Finite Field Diophantine Approximation Linear Fractional Transformation Continue Fraction Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BS1]
    Baum L. and Sweet M., Continued fractions of algebraic power series in characteristic 2, Annals of Mathematics, 103 (1976), 593–610.MathSciNetMATHCrossRefGoogle Scholar
  2. [BS2]
    Baum L. and Sweet M., Badly approximable power series in characteristic 2, Annals of Mathematics, 105 (1977), 573–580.MathSciNetMATHCrossRefGoogle Scholar
  3. [Ll]
    Lasjaunias A., A survey of diophantine approximation in fields of power series, Monatshefte für Mathematik, 130 (2000), 211–229.MathSciNetMATHCrossRefGoogle Scholar
  4. [L2]
    Lasjaunias A., Quartic power series in F3((T-1)) with bounded partial quotients, Acta Arithmetica, XCV. 1 (2000), 49–59.MathSciNetGoogle Scholar
  5. [L3]
    Lasjaunias A., Continued fractions for algebraic power series over a finite field, Finite Fields and their Applications, 5 (1999), 46–56.MathSciNetMATHCrossRefGoogle Scholar
  6. [LdM1]
    Lasjaunias A. and de Mathan B., Thue’s Theorem in positive characteristic, Journal für die reine und angewandte Mathematik, 473 (1996), 195–206.MathSciNetMATHCrossRefGoogle Scholar
  7. [LdM2]
    Lasjaunias A. and de Mathan B., Differential equations and diophantine approximation in positive characteristic, Monatshefte für Mathematik, 128 (1999), 1–6.MathSciNetMATHCrossRefGoogle Scholar
  8. [LR1]
    Lasjaunias A. and Ruch J-J., Algebraic and badly approximable power series over a finite field, Finite Fields and their Applications, 8 (2002), 91–107.MathSciNetMATHCrossRefGoogle Scholar
  9. [LR2]
    Lasjaunias A. and Ruch J-J., Flat power series over a finite field, Journal of Number Theory, to appear.Google Scholar
  10. [M]
    Mahler K., On a theorem of Liouville in fields of positive characteristic, Canadian Journal of Mathematics, 1 (1949), 397–400.MathSciNetMATHCrossRefGoogle Scholar
  11. [dM]
    de Mathan B. Approximation exponents for algebraic functions, Acta Arithmetica, LX. 4 (1992), 359–370.Google Scholar
  12. [MR]
    Mills W. and Robbins D., Continued fractions for certain algebraic power series, Journal of Number Theory, 23 (1986), 388–404.MathSciNetMATHCrossRefGoogle Scholar
  13. [S]
    Schmidt W., On Continued fractions and diophantine approximation in power series fields, Acta Arithmetica, XCV. 2 (2000), 139–165.Google Scholar
  14. [T]
    Thakur D., Diophantine approximation exponents and continued fractions for algebraic power series, Journal of Number Theory, 79 (1999), 284–291.MathSciNetMATHCrossRefGoogle Scholar
  15. [V]
    Voloch J-F., Diophantine approximation in positive characteristic, Periodica Mathematica Hungarica, 19. 3 (1988), 217–225.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alain Lasjaunias
    • 1
  1. 1.Université de Bordeaux ICNRS-UMR 5465Talence CedexFrance

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