On a theorem of Wirsing in Diophantine approximation
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Abstract:
Let $n$ and $d$ be integers with $1 \le d \le n-1$. Let $\xi$ be a real number which is not algebraic of degree at most $n$. We establish that there exist an effectively computable constant $c$, depending only on $\xi$ and on $n$, an integer $k$ with $1 \le k \le d$, and infinitely many integer polynomials $P(X)$ of degree $m$ at most equal to $n$ whose roots $\alpha _1, \ldots , \alpha _m$ can be numbered in such a way that \[ |(\xi - \alpha _1) \ldots (\xi - \alpha _k)| \le c H(P)^{-{d \over d+1}n - {1 \over d+1} - 1}. \] This extends a well-known result of Wirsing who dealt with the case $d=1$.References
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Additional Information
- Yann Bugeaud
- Affiliation: Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France
- Email: bugeaud@math.unistra.fr
- Received by editor(s): March 13, 2015
- Received by editor(s) in revised form: June 2, 2015
- Published electronically: October 2, 2015
- Communicated by: Matthew A. Papanikolas
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1905-1911
- MSC (2010): Primary 11J04
- DOI: https://doi.org/10.1090/proc/12879
- MathSciNet review: 3460153