We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1)) log n expected real zeros in terms of the degree n. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have n/ √ 3 + o(n) expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.

Abstract. Schur studied limits of the arithmetic means A n of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that lim sup n→∞ |A n | ≤ 1 − √ e/2. We show that A n → 0, and estimate the rate of convergence by generalizing the Erdős-Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk.Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for lim inf n→∞ A n was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line.Potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.

Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈ Mn(Z) satisfiesand the monic integer Chebyshev constant is then defined byE . This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.We compute t M (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. an interval whose endpoints are consecutive Farey fractions. This is characterized byThis should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

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.