Bounded approximation by polynomials with restricted zeros

Bounded Approximation by Polynomials Whose Zeros Lie on a Circleo

On the Zeros of Polynomials with Restricted Coefficients

It is proved that a polynomial p of the form

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.

Approximation by Polynomials with Coefficients

Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

We prove that the chromatic polynomial PG(q) of a finite graph G of maximal degree ∆ is free of zeros for |q| ≥ C∗(∆) with C∗(∆) = min 0<x<2 1 ∆ −1 (1 + x) x [2− (1 + x)] This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.