Multiplicity of zeros of polynomials

Multiplicity of Zeros and Discrete Orthogonal Polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero of some expansions ∑ aiFi(x), at a certain point c, depending on the chosen family {Fi}. The most important example is a polynomial with c = 1. It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the ze...

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.

Zeros of Interval Polynomials

Polynomial equations with perturbed coefficients arise in several areas of engineering sciences, for instance, in automatic control theory, dynamical systems, optimization and in control theory. For such equations, it is necessary to study their roots and to establish a priori estimates to define regions containing such roots. Attending to the fact that these equations can be seen as algebraic ...

Zeros of Fekete Polynomials

Zeros of Ramanujan Polynomials

In this paper, we investigate the properties of Ramanujan polynomials, a family of reciprocal polynomials with real coefficients originating from Ramanujan’s work. We begin by finding their number of real zeros, establishing a bound on their sizes, and determining their limiting values. Next, we prove that all nonreal zeros of Ramanujan polynomials lie on the unit circle, and are asymptotically...