When should a polynomial’s root nearest to a real number be real itself?
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- by A. Dubickas
- St. Petersburg Math. J. 25 (2014), 919-928
- DOI: https://doi.org/10.1090/S1061-0022-2014-01323-4
- Published electronically: September 8, 2014
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Abstract:
The conditions are studied under which the root of an integer polynomial nearest to a given real number $y$ is real. It is proved that if a polynomial $P \in \mathbb {Z}[x]$ of degree $d \geq 2$ satisfies $|P(y)| \ll 1/M(P)^{2d-3}$ for some real number $y$, where the implied constant depends on $d$ only, then the root of $P$ nearest to $y$ must be real. It is also shown that the exponent $2d-3$ is best possible for $d=2,3$ and that it cannot be replaced by a number smaller than $(2d-3)d/(2d-2)$ for each $d \geq 4$.References
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Bibliographic Information
- A. Dubickas
- Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
- Email: arturas.dubickas@mif.vu.lt
- Received by editor(s): October 4, 2012
- Published electronically: September 8, 2014
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 919-928
- MSC (2010): Primary 11C08, 11J04, 11R04
- DOI: https://doi.org/10.1090/S1061-0022-2014-01323-4
- MathSciNet review: 3234839