‎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.

1 We investigate the zeros of polynomial solutions to the differential-difference equation P n+1 (x) = A n (x)P ′ n (x) + B n (x)P n (x), n = 0, 1,. .. where A n and B n are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlac-ing. Our result holds for general classe...

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...

The resultant is an algebraic expression, computable in a finite number of arithmetic operations from the coefficients of two univariate polynomials, that vanishes if, and only if, the two polynomials have common zeros. The paper considers formal resultant for degree-deficient polynomials (polynomials whose actual degree is lower than their assumed degree). Some key properties of the resultant ...

Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha-theory to certify the results. Using this approach, we provide new examples of harmo...

with the initial values G0(x) = a and G1(x) = x + b. The polynomials Gn−1(1, 0;x) and Gn(2, 0;x)(n ≥ 1) are just the usual Fibonacci polynomials Fn(x) and the Lucas polynomials Ln(x), respectively. Our concern in this paper is to study some of the properties of the zeros of the Fibonacci-type polynomials Gn(a, b;x). The zeros of Fn(x) and Ln(x) have been given explicitly by V. E. Hoggatt, Jr. a...

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009New J. Phys. 11 023001) nowmake it possible to compute chromatic polynomia...

We extend results of Borcea and Brändén to quantitatively bound the movement of the zeros of Hurwitz stable polynomials under linear multiplier operators. For a multiplier operator T on polynomials of degree up to n, we show that the ratio between the maximal real part of the zeros of a Hurwitz stable polynomial f and that of its transform T [f ] is minimized for the polynomial (z + 1). For T o...

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, 2F1(−n, a;−n+ b; z), where n is a nonnegative integer. Owing to the connection between the classical Jacobi poly...