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

Exploring the properties of integer Chebyshev polynomials

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.