Free arrangements and relation spaces

Yuzvinsky [7] has shown that free arrangements are formal. In this note we define a more general class of arrangements which we callk-formal, and we show that free arrangements arek-formal. We close with an example which distinguishesk-formal arrangements from formal arrangements. The first author was supported in part by a U.S. Department of Education Fellowship. The second author was supported in part by the National Science Foundation.     

Free arrangements and rhombic tilings

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:

1. When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
2. When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?

Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.     

Free martingale polynomials

In this paper we investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.     

Small polynomials with integer coefficients

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

Linearization and connection coefficients of orthogonal polynomials

Let {P _n } _n ^=0/ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {P _n } _n ^=0/ are nonnegative then each of theP _n (x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsP _n can be expressed in terms ofQ _n with nonnegative coefficients, where {Q _n } _n ^=0/ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.     

Differential coefficients of orthogonal matrix polynomials

We find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given.     

On linearization coefficients of Jacobi polynomials

This article deals with the problem of finding closed analytical formulae for generalized linearization coefficients for Jacobi polynomials. By considering some special cases, we obtain a reduction formula using for this purpose symbolic computation, in particular Zeilberger’s and Petkovsek’s algorithms.     

Zolotarev polynomials and reduction of Shabat polynomials into a positive characteristic

The paper is focused on the study of Shabat polynomials over fields of different characteristics and their deformation into polynomials with three critical values. Using this deformation, we obtain prime numbers of bad reduction for Shabat polynomials corresponding to trees of diameter 4.     

Approximation by polynomials with small coefficients

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 150–153, January, 1995.