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When should a polynomial's root nearest to a real number be real itself?


Author: A. Dubickas
Original publication: Algebra i Analiz, tom 25 (2013), nomer 6.
Journal: St. Petersburg Math. J. 25 (2014), 919-928
MSC (2010): Primary 11C08, 11J04, 11R04
Published electronically: September 8, 2014
MathSciNet review: 3234839
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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 $ \vert P(y)\vert \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$.


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Additional Information

A. Dubickas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Email: arturas.dubickas@mif.vu.lt

DOI: https://doi.org/10.1090/S1061-0022-2014-01323-4
Keywords: Polynomial root separation, real roots, Mahler's measure, discriminant
Received by editor(s): October 4, 2012
Published electronically: September 8, 2014
Article copyright: © Copyright 2014 American Mathematical Society