대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제49권1호
  • /
  • Pages.127-133
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  • 2012
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))

  • Hbaib, Mohamed (Departement de Mathematiques Faculte des Sciences de Sfax)
  • 투고 : 2010.09.01
  • 발행 : 2012.01.31
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초록

It is well known that if the ${\beta}$-expansion of any nonnegative integer is finite, then ${\beta}$ is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the ${\beta}$-expansion of the polynomial part of ${\beta}$ is finite if and only if ${\beta}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\mathbb{F}_q((x^{-1}))$. Finally we show that if the base ${\beta}$ is a Pisot series, then there is a bound of the length of the fractional part of ${\beta}$-expansion of any polynomial P in $\mathbb{F}_q[x]$.

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참고문헌

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  6. J. Neukirch, Algebraic Number Theory, Translated from the 1992 German original and with a note by Norbert Schappacher.With a foreword by G. Harder.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999.
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피인용 문헌

  1. Continued $$\beta $$ β -fractions with formal power series over finite fields vol.39, pp.1, 2016, https://doi.org/10.1007/s11139-015-9725-5