A sufficient condition for a polynomial to be stable

分析理论与应用 英文刊, 2009

Let P(z) = a n z n + a n−1 z n−1 + ··· + a 0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coefficients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).

Illinois Journal of Mathematics, 1997

Linear Algebra and its Applications, 2011

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.

Pacific Journal of Mathematics, 1985

2012

Let P(z) be a polynomial of degree n with real or complex coefficients . The aim of this paper is to obtain a ring shaped region containing all the zeros of P(z). Our results not only generalize some known results but also a variety of interesting results can be deduced from them.

Kragujevac Journal of Mathematics, 2022

If P(z) is a polynomial of degree n, then for a subclass of polynomials, Dalal and Govil [7] compared the bounds, containing all the zeros, for two different results with two different real sequences λk > 0, Pn k=1 λk = 1. In this paper, we prove a more general result, by which one can compare the bounds of two different results with the same sequence of real or complex λk, Pn k=0 ♣λk♣ ≤ 1. A variety of other results have been extended in this direction, which in particular include several known extensions and generalizations of a classical result of Cauchy [4], from this result by a fairly uniform manner.

Acta Universitatis Sapientiae: Mathematica, 2022

2008

The notes contain a streamlined account on stability of univariate polynomials and related problems

2012

In this paper, we establish some relations between the zeros and coefficients of a polynomial and thereby prove a few results concerning stable polynomials.

Journal of Pure and Applied Algebra, 2021

Given a proper cone K ⊆ R n , a multivariate polynomial f ∈ C[z] = C[z 1 ,. .. , z n ] is called K-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of K. If K is the non-negative orthant, then Kstability specializes to the usual notion of stability of polynomials. We study conditions and certificates for the K-stability of a given polynomial f , especially for the case of determinantal polynomials as well as for quadratic polynomials. A particular focus is on psd-stability. For cones K with a spectrahedral representation, we construct a semidefinite feasibility problem, which, in the case of feasibility, certifies K-stability of f. This reduction to a semidefinite problem builds upon techniques from the connection of containment of spectrahedra and positive maps. In the case of psd-stability, if the criterion is satisfied, we can explicitly construct a determinantal representation of the given polynomial. We also show that under certain conditions, for a K-stable polynomial f , the criterion is at least fulfilled for some scaled version of K. Theorem 2.3. For a homogeneous polynomial f ∈ R[z], the following are equivalent. (1) f is K-stable. (2) I(f) ∩ int K = ∅. (3) f is hyperbolic w.r.t. every point in int K. By [21], the hyperbolicity cones of a homogeneous polynomial f coincide with the components of I(f) c , where I(f) c denotes the complement of I(f). This implies: Corollary 2.4. A hyperbolic polynomial f ∈ R[z] is K-stable if and only if int K ⊆ C(e) for some hyperbolicity direction e of f. Proof. This follows from the observation that a hyperbolic polynomial f ∈ R[z] is K-stable if and only if int K ⊆ I(f) c. It is shown in [21] that the number of hyperbolicity cones of a homogeneous polynomial f ∈ R[z] is at most 2 d for d ≤ n and at most 2 n−1 k=0 d−1 k for d > n.

International Journal of Pure and Apllied Mathematics, 2013

Let P (z) be a polynomial of degree n with decreasing coefficients. Then all its zeros lie in |z| ≤ 1. In this paper we present some generalizations of this result and a refinement of a classical bounds.

Michigan Mathematical Journal, 2001

For z 0 ∈ C and r > 0, let D(z 0 , r) := {z ∈ C : |z − z 0 | < r}. In this paper we show that a polynomial p of the form (*) p(x) = n j=0 a j x j , |a 0 | = 1 , |a j | ≤ 1 , a j ∈ C , has at most (c 1 /α) log(1/α) zeros in the disk D(0, 1 − α) for every α ∈ (0, 1), where c 1 > 0 is an absolute constant. This is a simple consequence of Jensen's formula. However it is not so simple to show that this estimate for the number of zeros in D(0, 1 − α) is sharp. We will present two examples to show the existence of polynomials pα (α ∈ (0, 1)) of the form (*) (with a suitable n ∈ N depending on α) with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant). In fact, we will show the existence of such polynomials from much smaller classes with more restrictions on the coefficients. Our first example has probabilistic background and shows the existence of polynomials pα (α ∈ (0, 1)) with complex coefficients of modulus exactly 1 and with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant). Our second example is constructive and defines polynomials pα (α ∈ (0, 1)) with real coefficients of modulus at most 1, with constant term 1, and with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant).

Journal of Mathematical Analysis and Applications, 1980

Let P(z) be a polynomial of degree n with real or complex coefficients, In this paper we obtain a ring shaped region containing all the zeros of P( z), Our results include, as special cases, several known extensions of Enestrom-Kakeya theorem on the zeros of a polynomiaL We shall also obtain zero free regions for certain class of analytic functions. n THEOREMB. LetP(z) = L aki ¥=-°be apolynomial with complex coefficients k=O such that I arg ak -131 :::; a :::;~, k = 0, l, ... , n for some 13, and Mathematics subject classification (1991): 30CIO,30CI5.

Mathematics, 2019

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

Linear Algebra and its Applications, 2011