ИСТИНА

An algebraic polynomial is expanding if all its roots are outside the unit circle of the complex plane. The problem of classification of integer expanding polynomials arises naturally in many applications. We address two of them: self-similar space tilings and the Euler binary partition functions. Tiles are used in the construction of multivariate Haar bases and of wavelets and in this connection they have been studied in detail by Lagarias, Wang, Grochenig, Daubechies, Cohen, Wojtaszczyk, etc. The total number of tiles of dimension n is expressed in terms of the number of monic integer expanding polynomials of degree n with a given constant term. The asymptotics of this value is estimated from above by Mahler’s measure and by results of Dubickas and Konyagin, while the lower bounds are obtained by constructing new series of such polynomials. The second application is from the combinatorial number theory. The Euler binary partition function is the total number of expansions of a given number as a sum of powers of two (with repetitions permitted). The asymptotic growth of this function is estimated by Newman polynomials. For characterizing of the regular power growth one introduces the concept of k-cyclotomic polynomials, which lead to the problems of spline approximations. A part of the results are obtained in collaboration with Y.Wang (HKUST, Hon Kong) and with T.Zaitseva (MSU, Russia).