We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random p...

If P(z) be a polynomial of degree n with decreasing coefficients, then all its zeros lie in |z| ≤ 1. In this paper we find new classical results concerning Enestrom-Kakeya theorem and related analytic functions. Besides several consequences, our results improve the bounds by relaxing and weakening the hypothesis in some cases.

the aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear irreducible characters in the character tables. we find that they are exactly a$_{5}$, l$_{2}(7)$ and a$_{6}$.

Generalized nonnegative polynomials were studied in a sequence of recent papers [4], [6], [7], [8], and [9]. A number of well-known inequalities in approximation theory were extended to them, by utilizing the generalized degree in place of the ordinary one. Our motivation was to find tools to examine systems of orthogonal polynomials simultaneously, associated with generalized Jacobi, or at lea...

The notions separability and normality are related to this characterisation. In the case of skew fields polynomials often have infinitely many zeros, so a different way of counting zeros as distinct is needed. The well-known theorem of Gordon and Motzkin [Z] states that a polynomial of degree n has zeros in at most n conjugacy classes. This suggests one should count zeros of a polynomial by the...

We prove that there exists an absolute constant c > 0 such that |p′(y)| ≤ c min { n(k + 1), ( n(k + 1) 1 − y2 )1/2} max −1≤x≤1 |p(x)|, −1 ≤ y ≤ 1 for every real algebraic polynomial of degree at most n having at most k zeros in the open unit disk {z ∈ C : |z| < 1}. This inequality, which has been conjectured for at least a decade, improves and generalizes several earlier results. Up to the mult...

Abstract We note that the recent polynomial proofs of spherical and complex plank covering problems by Zhao Ortega-Moreno give some general information on zeros real polynomials restricted to unit sphere. As a corollary these results, we establish several generalizations celebrated Bang theorem. prove tight analog theorem for Euclidean ball an even stronger version projective space. Specificall...

Abstract. The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ(s), its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of ξ(s) under the assumption that ...

In the noncommutative case also a version of (1) can be proved; see for instance [S, Proposition A.33. In that case, in general, such an N is not finitely generated over K. What (2) means for the noncommutative case depends on how one defines the phrase “constructing by repeatedly adding zeros of p until p has a complete set of zeros.” The adding of zeros may be done by forming field coproducts...

Polynomials with perturbed coefficients, which can be regarded as interval polynomials, are very common in the area of scientific computing due to floating point operations in a computer environment. In this paper, the zeros of interval polynomials are investigated. We show that, for a degree n interval polynomial, the number of interval zeros is at most n and the number of complex block zeros ...