Small polynomials with integer coefficients

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 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 integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for integer Chebyshev constant. Moreover, all the mentioned bounds can be found numerically, by using various extremal point techniques, such as 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. 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15.

A factor of integer polynomials with minimal integrals

“…Furthermore, Pritsker [11,Theorem 3.1] showed that C ∈ ]0.85991, 0.86441[, and this is the best estimate of C known to date.…”

The Multivariate Integer Chebyshev Problem

“…Furthermore, it was shown in [20] that, for the circle L 1/n = {z : |nz − 1| = 1/n}, n ∈ N, n ≥ 2, we have t Z (L 1/n ) = 1/n and t C (L 1/n ) = 1/n 2 . Hence equality holds in (1.9) in this case.…”

“…They heavily depend on the arithmetic properties of endpoints of the interval. Different methods based on weighted potential theory are employed in [20]. We note that the exact values of t Z are not known for any segment of length less than 4.…”

“…Following the univariate case [13,18,6,12,20], it is possible to give a lower bound for the multivariate integer Chebyshev constant by using the leading coefficients of minimal polynomials for algebraic numbers in the set.…”

Applications of Integer Semi-Infinite Programing to the Integer Chebyshev Problem

“…At the time no non-trivial lower bound for ǫ was determined. Pritsker showed in [19], by means of weighted potential theory, that t Z ([0, 1]) ≥ 0.4213. Generalizations of these Gorshkov-Wirsing polynomails were considered in [15].Given the submultiplicative nature of t Z,n (I) we have t Z (I) ≤ t Z,n (I) for all n. This gives a simple method to find an upper bound for t Z (I); find large degree polynomials with small supremum norm.…”