Real Polynomial Research Articles

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.

Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps

A real univariate polynomial is called hyperbolic or stable if all its roots are real. We search for hyperbolic polynomials of two and three degrees by using the Wronskian map W and a dual map to W called Leibniz, since it involves the classical Leibniz rule for the derivative of a product of functions. In addition to hyperbolicity, we use these two methods to search for a class of polynomials introduced by the first author and now called weak Euclidean.

On Descartes' rule of signs for hyperbolic polynomials

We consider univariate real polynomials with all real roots and with two sign changes in the sequence of their coefficients which are all non-vanishing. Assume that one of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree \(d\) polynomials have 2 positive and \(d-2\) negative roots. We consider the possible sequences of the moduli of their roots on the real positive half-axis. When these moduli are distinct, we give the exhaustive answer to the question which positions can the moduli of the two positive roots occupy.

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree [Formula: see text] and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces [Formula: see text] of real monic univariate polynomials of degree [Formula: see text] whose real divisors avoid sequences of root multiplicities, taken from a given poset [Formula: see text] of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of [Formula: see text] and of some related topological spaces. We find explicit presentations for the groups [Formula: see text] in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in [Formula: see text]. We also show that [Formula: see text] stabilizes for [Formula: see text] large. The mechanism that generates [Formula: see text] has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree [Formula: see text] polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups [Formula: see text] admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder [Formula: see text], whose images avoid the tangency patterns from [Formula: see text] with respect to the generators of the cylinder.

Real factorization of positive semidefinite matrix polynomials

Suppose Q(x) is a real n×n regular symmetric positive semidefinite matrix polynomial. Then it can be factored asQ(x)=G(x)TG(x), where G(x) is a real n×n matrix polynomial with degree half that of Q(x) if and only if det⁡(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equationXSX−XR+RTX+P=0, where P,R,S are real n×n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization.

Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

The Probabilistic Method in Real Singularity Theory

We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.

Bifurcating Limit Cycles with a Perturbation of Systems Composed of Piecewise Smooth Differential Equations Consisting of Four Regions

Systems composed of piecewise smooth differential (PSD) mappings have quantitatively been searched for answers to a substantial issue of limit cycle (LC) bifurcations. In this paper, LC numbers (LCNs) of a PSD system (PSDS) consisting of four regions are dealt with. A Melnikov mapping whose order is one is implicitly obtained by finding its originators when the system is perturbed under any nth degree of real polynomials. Then, the approach employing the Picard–Fuchs mapping is utilized to attain a higher boundary of bifurcation LCNs of systems composed of PSD functions with a global center. The method we used could be implemented to examine the problems related to the LC of other PSDS.

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

Computing the Non-properness Set of Real Polynomial Maps in the Plane

AbstractWe introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set, is a subset of $$\mathbb {K}^2$$ K 2 , where a dominant polynomial map $$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2$$ f : K 2 → K 2 is not proper; $$\mathbb {K}$$ K could be either $$\mathbb {C}$$ C or $$\mathbb {R}$$ R . Unlike all the previously known approaches we make no assumptions on f whenever $$\mathbb {K} = \mathbb {R}$$ K = R ; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in maple.

EXPONENTIAL POLYNOMIALS AND STRATIFICATION IN THE THEORY OF ANALYTIC INEQUALITIES

This paper considers MEP - Mixed Exponential Polynomials as one class of real exponential polynomials. We introduce a method for proving the positivity of MEP inequalities over positive intervals using the Maclaurin series to approximate the exponential function precisely. Additionally, we discuss the relation between MEPs and stratified families of functions from [1] through two applications, referring to inequalities from papers [2] and [3].

Amalgamation of real zero polynomials

With this article, we hope to launch the investigation of what we call the Real Zero Amalgamation Problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension of the first one. The Real Zero Amalgamation Problem asks when two (multivariate real) polynomials have a common extension (called amalgam) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turzík. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turzík. This would imply a very weak form of the Generalized Lax Conjecture.

Bipartite perfect matching as a real polynomial

Bipartite perfect matching as a real polynomial

Expressive curves

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.

On stability of spanning tree degree enumerators

We show that the spanning tree degree enumerator polynomial of a connected graph G is a real stable polynomial if and only if G is distance-hereditary.

Estimating the Number of Zeros of Abelian Integrals for the Perturbed Cubic Z4-Equivariant Planar Hamiltonian System

We analyze the dynamics of a class of [Formula: see text]-equivariant Hamiltonian systems of the form [Formula: see text], where [Formula: see text] is complex, the time [Formula: see text] is real, while [Formula: see text] and [Formula: see text] are real parameters. The topological phase portraits with at least one center are given. The finite generators of Abelian integral [Formula: see text] are obtained, where [Formula: see text] is a family of closed ovals defined by [Formula: see text] [Formula: see text], [Formula: see text] is the open interval on which [Formula: see text] is defined, [Formula: see text], [Formula: see text] are real polynomials in [Formula: see text] and [Formula: see text] with degree [Formula: see text]. We give an estimation of the number of isolated zeros of the corresponding Abelian integral by using its algebraic structure. We show that for the given polynomials [Formula: see text] and [Formula: see text] in [Formula: see text] and [Formula: see text] with degree [Formula: see text], the number of the limit cycles of the perturbed [Formula: see text]-equivariant Hamiltonian system does not exceed [Formula: see text] (taking into account the multiplicity).

Beyond Descartes’ rule of signs

We consider real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is.

Lower bounds on the Hausdorff dimension of some Julia sets

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial . In addition to that, we construct a piecewise constant function on that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials with . Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial , is a C 1-smooth function of the real parameter c on the interval .

Persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno’s conditions

This paper proves the persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno non-degeneracy conditions, whose frequency vector is a small dilation of the prescribed one. The proof is based on the stability of real roots of approximating real odd-order polynomials.

On C. Michel’s hypothesis about the modulus of typically real polynomials

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szegő, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new results on extremal properties of typically real polynomials have been obtained. Fundamental work in this direction is due to M. Brandt, who found a novel way of solving extremal problems. In particular, he solved C. Michel’s problem of estimating the modulus of a typically real polynomial of odd degree. On the other hand, D. K. Dimitrov showed the efficiency of Fejér’s method for solving the Rogosinski–Szegő problems. In this article, we completely solve Michel’s problem by using Fejér’s method.