Criterion For Polynomials Research Articles

A Family of Fractal-Based Irreducible Polynomials

Summary We provide some fractal-based irreducibility criteria for integer polynomials. The results are obtained using the self-similar pattern of Pascal’s triangle modulo two and by applying a restatement of the Schönemann–Eisenstein’s irreducibility criterion.

New negative-determination conditions for cubic polynomials with applications to time-varying delay systems

This paper studies the stability of time delay systems. The Lyapunov-Krasovskii functional (LKF) method is used for our study, in which a novel negative-determination criterion for cubic polynomials is proposed. An improved stability criterion of time delay system is obtained by the new method. The effectiveness of the proposed method is verified by some numerical examples and reduced conservativeness can be obtained.

Factors of HOMFLY polynomials

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.

Minimality of rational knots C(2n + 1,2m,2)

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots [Formula: see text] in the Conway notation, where [Formula: see text] and [Formula: see text] are integers. When [Formula: see text], we show that the nonabelian [Formula: see text]-character variety of [Formula: see text] is irreducible and therefore [Formula: see text] is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.

Scale fragilities in localized consensus dynamics

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher (n≥2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents (n≥3), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus (n=2) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh–Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue.

A reduced stability criterion for Schur stable polynomials

A reduced stability criterion for Schur stable polynomials

Stieltjes Property of Quasi-Stable Matrix Polynomials

In this paper, basing on the theory of matricial Hamburger moment problems, we establish the intrinsic connections between the quasi-stability of a monic or comonic matrix polynomial and the Stieltjes property of a rational matrix-valued function built from the even–odd split of the original matrix polynomial. As applications of these connections, we obtain some new criteria for quasi-stable matrix polynomials and Hurwitz stable matrix polynomials, respectively.

Split absolutely irreducible integer-valued polynomials over discrete valuation domains

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R,M) with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements.We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call balanced, characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of M. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible.

Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with $ s $-convexity

<abstract><p>In this paper, we generate some non-classical variants of Julia and Mandelbrot sets, utilizing the Jungck-Ishikawa fixed point iteration system equipped with $ s $-convexity. We establish a novel escape criterion for complex polynomials of a higher degree of the form $ z^n + az^2 -bz + c $, where $ a, \; b $ and $ c $ are complex numbers and furnish some graphical illustrations of the generated complex fractals. In the sequel, we discuss the errors committed by the majority of researchers in developing the escape criterion utilizing distinct fixed point iterations equipped with $ s $-convexity. We conclude the paper by examining variation in images and the impact of parameters on the deviation of dynamics, color and appearance of fractals. It is fascinating to notice that some of our fractals represent the traditional Kachhi Thread Works found in the Kutch district of Gujarat (India) which is useful in the Textile Industry.</p></abstract>

Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

For a real polynomial f we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of f, and vertices expressed in terms of the coefficients of f. We also describe zero-free lens-shaped regions for f that are associated to the zero-free sectors of the reciprocal of f. As an application, we use these zero-free regions to obtain irreducibility criteria for polynomials with integer coefficients that take a prime or a prime power value.

Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials f(X) whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae a and b, with ratio of the distances to these points depending on the canonical decomposition of f(a) and f(b). In particular, we obtain irreducibility criteria for the case where f(a) and f(b) have few prime factors, and f is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

Finding Compositional Inverses of Permutations From the AGW Criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.

Positivity of Turán determinants for orthogonal polynomials II

The polynomials pn orthogonal on the interval [−1,1], normalized by pn(1)=1, satisfy Turán’s inequality if pn2(x)−pn−1(x)pn+1(x)≥0 for n≥1 and for all x in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán’s inequality. This extends essentially the results of Szwarc(1998). In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

105.06 The Hermite-Sylvester criterion for real-rooted polynomials

105.06 The Hermite-Sylvester criterion for real-rooted polynomials

On the Abhyankar–Moh irreducibility criterion for quasi-ordinary polynomials

Let $f$ and $g$ be Weierstrass polynomials with coefficients in the ring of formal power series over an algebraically closed field of characteristic zero. Assume that $f$ is irreducible and quasi-ordinary. We show that if the degree of $g$ is small enough

On a Dual to the Properties of Hurwitz Polynomials I

In this work we develop necessary and sufficient conditions for describing the family of anti-Hurwitz polynomials, introduced by Vergara-Hermosilla et al. in [1]. Specifically, we studied a dual version of the Theorem of Routh-Hurwitz and present explicit criteria for polynomials of low order and derivatives. Another contribution of this work is establishing a dual version of the Hermite-Biehler Theorem. To this aim, we give extensions of the boundary crossing Theorems and a zero exclusion principle for anti-Hurwitz polynomials.

Comparative analysis of stability criteria for polynomials

The article provides a comparative analysis of two stability criteria for polynomials. They do not require obtaining any information about the roots, but allow us to judge stability directly by the coefficients of the polynomial. On specific numerical examples, the features of their application are considered. The article also provides symbolic expressions of the relationship between the coefficients of sixth and fourth degree polynomials for cases where there is a linear or definite nonlinear relationship between the roots of these polynomials. In these cases, an algebraic equation of the sixth degree can be solved in radicals.

On generalization of classical Hurwitz stability criteria for matrix polynomials

In this paper we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion.

Computing Closest Stable Nonnegative Matrix

The problem of finding the closest stable matrix for a dynamical system has many applications. It is studied for both continuous and discrete-time systems and the corresponding optimization problems are formulated for various matrix norms. As a rule, nonconvexity of these formulations does not allow finding their global solutions. In this paper, we analyze positive discrete-time systems. They also suffer from nonconvexity of the stability region, and the problem in the Frobenius norm or in the Euclidean norm remains hard for them. However, it turns out that for certain polyhedral norms, the situation is much better. We show that for the distances measured in the max-norm, we can find an exact solution of the corresponding nonconvex projection problems in polynomial time. For the distance measured in the operator $\ell_{\infty}$-norm or $\ell_{1}$-norm, the exact solution is also efficiently found. To this end, we develop a modification of the recently introduced spectral simplex method. On the other hand, for all these three norms, we obtain exact descriptions of the region of stability around a given stable matrix. In the case of the max-norm, this can be seen as an extension onto the class of nonnegative matrices, the Kharitonov theorem, providing a stability criterion for polynomials with interval coefficients [V. L Kharitonov, Differ. Uravn., 14 (1978), pp. 2086--2088; K. Panneerselvam and R. Ayyagari, Internat. J. Control Sci. Engrg., 3 (2013), pp. 81--85]. For practical implementation of our technique, we developed a new method for approximating the maximal eigenvalue of a nonnegative matrix. It combines the local quadratic rate of convergence with polynomial-time global performance guarantees.

A Useful Irreducibility Result for Integer Polynomials

We give a useful irreducibility criterion for integer polynomials by means of an elementary approach. We also illustrate the result by providing examples.