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Small polynomials with integer coefficients

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Abstract

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates.

Since it is rarely possible to obtain an exact value of the integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for the integer Chebyshev constant. Moreover, all the bounds mentioned can be found numerically by using various extremal point techniques, such as the weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials.

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Authors and Affiliations

  1. Department of Mathematics 401 Mathematical Sciences, Oklahoma State University, 74078-1058, Stillwater, OK, USA

    Igor E. Pritsker

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  1. Igor E. Pritsker
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Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.

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Pritsker, I.E. Small polynomials with integer coefficients. J. Anal. Math. 96, 151–190 (2005). https://doi.org/10.1007/BF02787827

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  • DOI: https://doi.org/10.1007/BF02787827

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