Analysis In Different Phases, Formulas and Conditions of Integer Coefficients Polynomials

Suman Lata

Analysis In Different Phases, Formulas and Conditions of Integer Coefficients Polynomials

Exploring the properties of integer Chebyshev polynomials

by Suman Lata*,

- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659

Volume 2, Issue No. 1, Aug 2011, Pages 0 - 0 (0)

Published by: Ignited Minds Journals

ABSTRACT

We study the problem of minimizingthe supremum norm, on a segment of the real line or on a compact set in theplane, by polynomials with integer coefficients. The extremal polynomials arenaturally called integer Chebyshev polynomials. Their factors, zerodistribution and asymptotics are the main subjects of this paper. Inparticular, we show that the integer Chebyshev polynomials for any infinitesubset of the real line must have infinitely many distinct factors, whichanswers a question of Borwein and Erdelyi. Furthermore, it is proved that theaccumulation set for their zeros must be of positive capacity in this case. Wealso find the first nontrivial examples of explicit integer Chebyshev constantsfor certain classes of lemniscates.

KEYWORD

analysis, phases, formulas, conditions, integer coefficients polynomials, minimizing, supremum norm, real line, compact set, polynomials, integer Chebyshev polynomials, factors, zero distribution, asymptotics, infinitely many distinct factors, accumulation set, positive capacity, explicit integer Chebyshev constants, lemniscates

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