For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

2000 Mathematics Subject Classification: 12D10.In the present paper we consider degree 6 hyperbolic polynomials (HPs) in one variable (i.e. real and with all roots real). We are interested in such HPs whose number of equalities between roots of the polynomial and/or its derivatives is higher than expected. We give the complete study of the four families of such degree 6 even HPs and also of HPs which are primitives of degree 5 HPs.Research partially supported by research project 20682 for cooperation...

MSC 2010: 30C10The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of several well-known results...

We study Mellin transforms $\widehat{N}\left(s\right)={\int }_{1-}^{\infty }{x}^{-s}dN\left(x\right)$ for which $N\left(x\right)-x$ is periodic with period $1$ in order to investigate ‘flows’ of such functions to Riemann’s $\zeta \left(s\right)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N\left(x\right)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac{1}{2}$.We investigate a particular flow of such functions ${\left\{\widehat{N_{\lambda }}\right\}}_{\lambda \ge 1}$ which converges locally uniformly to $\zeta \left(s\right)$ as $\lambda \to 1$, and show that they exhibit features similar to $\zeta \left(s\right)$. For example, $\widehat{N_{\lambda }}\left(s\right)$...

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.