On the zeros of polynomials

In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary type polynomials.     

On the zeros of exponential polynomials

Suppose r = (r₁, …, r_M), r_j ? 0, γ_kj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γ_k · r = ∑_jγ_kjr_j. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑_{j = 1}^Na_je^{?λγ_j · r}, where each a_j is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑_{k = 1}^MA_kx(t ? r_k) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.     

On the zeros of certain polynomials

We prove that certain naturally arising polynomials have all of their roots on a vertical line.     

On the zeros of subrange Jacobi polynomials

We introduce and study a geometric modification of the Douglas–Rachford method called the Circumcentered–Douglas–Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas–Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered–Douglas–Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas–Rachford method. We also present a preliminary discussion on the Circumcentered–Douglas–Rachford method applied to the many set case and to examples featuring non-affine convex sets.     

On the zeros of certain self-reciprocal polynomials

We investigate the distribution of zeros around the unit circle of real self-reciprocal polynomials of even degrees with five terms whose absolute values of middle coefficients equal the sum of all other coefficients. Furthermore, it also give a new inequality and other Eneström-Kakeya types of results as by-products of this investigation.     

On multiple orthogonal polynomials for three Meixner measures

Multiple orthogonal polynomials for three discrete Meixner measures with identical exponential decay at infinity are studied. These polynomials are the denominators of the type II Hermite–Padé approximants to some hypergeometric functions. The limit distribution of zeros of such polynomials scaled in a certain way is described in terms of equilibrium logarithmic potentials and in terms of algebraic curves.     

On the fourth-order difference equation for the associated Meixner polynomials

Three equivalent forms of the fourth-order difference equation obeyed by the associated Meixner polynomials (with a nonnegative real association parameter) are derived from a refinement of a recent result due to Letessier et al. (1996).     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

Asymptotics for multiple Meixner polynomials

We study the asymptotic behavior of multiple Meixner polynomials of first and second kind, respectively [6]. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.     

On the distribution of simple zeros of polynomials

Erd s and Turán discussed in (Ann. of Math. 41 (1940), 162–173; 51 (1950), 105–119) the distribution of the zeros of monic polynomials if their Chebyshev norm on [−1, 1] or on the unit disk is known. We sharpen this result to the case that all zeros of the polynomials are simple. As applications, estimates for the distribution of the zeros of orthogonal polynomials and the distribution of the alternation points in Chebyshev polynomial approximation are given. This last result sharpens a well-known error bound of Kadec (Amer. Math. Soc. Transl. 26 (1963), 231–234).     

Some characteristic property of the Meixner polynomials

A new characterization of the Meixner polynomials is established. It is based on the solution of a problem related to a previous result concerning the Laguerre polynomials. Solutions of analogous problems provide characteristic properties of the Laguerre and Hermite polynomials. These properties, which are derived from the two-variable polynomials, generalize, in turn, the previous ones.     

On permanents and the zeros of rook polynomials

The concept of rook polynomial of a “chessboard” may be generalized to the rook polynomial of an arbitrary rectangular matrix. A conjecture that the rook polynomials of “chessboards” have only real zeros is thus carried over to the rook polynomials of nonnegative matrices. This paper proves these conjectures, and establishes interlacing properties for the zeros of the rook polynomials of a positive matrix and the matrix obtained by striking any one row or any one column.     

On the zeros of complex Van Vleck polynomials

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials, have been studied extensively in the case of real number parameters. In the complex case, relatively little is known. Numerical investigations of the location of the zeros of the Stieltjes and Van Vleck polynomials in special cases reveal intriguing patterns in the complex case, suggestive of a deeper structure. In this article we report on these investigations, with the main result being a proof of a theorem confirming that the zeros of the Van Vleck polynomials lie on special line segments in the case of the complex generalized Lamé equation having three free parameters. Furthermore, as a result of this proposition, we are able to obtain in this case a strengthening of a classical result of Heine on the number of possible Van Vleck polynomials associated with a given Stieltjes polynomial.     

On relations between zeros of Galois polynomials

A conjecture ofH. Kleiman says that over certain fields a Galois equation of degree 3 is uniquely determined by its root polynomials. We prove this conjecture for prime degrees 3 and a somewhat smaller class of fields than Kleiman's. In this situation, the ideal of all relations between zeros of the equation has a basis containing root polynomials only, not the equation itself. Giving a large class of counterexamples of degree 4, we disprove Kleiman's conjecture in general.     

On the perturbation of the zeros of complex polynomials

We investigate the deviation of the zeros of a polynomial. Thedeviation grows with the nth root of the perturbation of thepolynomial, where n is the degree of the polynomial, and thetask is to determine the factor in front of this root. We improveupon earlier results by Ostrowski and Schönhage and givesharp estimates. Applications are given in estimating the differencesof root-radii, which is important for the computation of thezeros of polynomials.