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Статьи
When should a polynomial's root nearest to a real number be real itself?
A. Dubickas Department of Mathematics and Informatics, Vilnius University, Naugarduko, 24, Vilnius LT-03225, Lithuania
Аннотация:
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 $d\geq2$ satisfies $|P(y)|\ll1/M(P)^{2d-3}$ 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 $d\geq4$.
Ключевые слова:
polynomial root separation, real roots, Mahler's measure, discriminant.
Поступила в редакцию: 04.10.2012
Образец цитирования:
A. Dubickas, “When should a polynomial's root nearest to a real number be real itself?”, Алгебра и анализ, 25:6 (2013), 37–49; St. Petersburg Math. J., 25:6 (2014), 919–928
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/aa1362 https://www.mathnet.ru/rus/aa/v25/i6/p37
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