Roots Of Polynomial Research Articles

On the Degree of the GCD of Random Polynomials over a Finite Field

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .

Algorithms and Programs for Calculating the Roots of Polynomial of One or Two Variables

Algorithms and Programs for Calculating the Roots of Polynomial of One or Two Variables

Lengths of roots of polynomials in a Hahn field

Lengths of roots of polynomials in a Hahn field

House of algebraic integers symmetric about the unit circle

We give a Schinzel-Zassenhaus-type lower bound for the maximum modulus of roots of a monic integer polynomial with all roots symmetric with respect to the unit circle. Our results extend a recent work of Dimitrov, who proved the general Schinzel-Zassenhaus conjecture by using the Pólya rationality theorem for a power series with integer coefficients, and some estimates for logarithmic capacity (transfinite diameter) of sets. We use an enhancement of Pólya's result obtained by Robinson, which involves Laurent-type rational functions with small supremum norms, thereby replacing the logarithmic capacity with a smaller quantity. This smaller quantity is expressed via a weighted Chebyshev constant for the set associated with Dimitrov's function used in Robinson's rationality theorem. Our lower bound for the house confirms a conjecture of Boyd.

The roots of polynomials and the operator $$\Delta _i^3$$ on the Hahn sequence space h

The roots of polynomials and the operator $$\Delta _i^3$$ on the Hahn sequence space h

Deep Learning Based Uncertainty Analysis in Computational Micromechanics of Composite Materials

Design of new materials is quite a difficult task owing to various time and length scales and affiliated uncertainties. The major challenge is to include all these in a conventional model. Hyperparameter models in machine learning can be used to overcome these issues. In this paper, an artificial neural network (ANN) model is developed to estimate the effective elastic parameters of unidirectional fiber reinforced composites using representative volume elements (RVE) considering uncertainty in the fiber diameter. The diameter probability distribution is constructed from the acquired gray images by employing image processing operations. The generalized Polynomial Chaos (gPC) expansion is then used to represent the distribution as a random input parameter for finite element analysis, from where the effective parameters are realized. Similarly, the outputs of the FE model, i.e., elastic parameters, are approximated by gPC expansions having unknown deterministic coefficients and random orthogonal Hermite polynomials. A set of collocation points are generated from roots of the random polynomials; from there, the unknown coefficients are estimated. The realization samples are utilized to train an ANN algorithm based on supervised deep learning. The developed ANN model is later tested and validated for a new sample set of data. It is shown that the ANN model with few hidden layers and neurons has a high accuracy for estimation of the elastic parameters directly from the information on the distribution of fiber diameters.

Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials W_n(z) = sum _{k=0}^{n-1} xi _k z^k concentrate around the unit circle when the coefficients of W_n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as nrightarrow infty if and only if {mathbb {E}}[log ( 1+ |xi _0|)]<infty . Under the condition {mathbb {E}}[xi ^2_0]<infty , we show that there exists an annulus of width {text {O}}(n^{-2}(log n)^{-3}) around the unit circle which is free of roots with probability 1-{text {O}}({(log n)^{-{1}/{2}}}). The proof relies on small ball probability inequalities and the least common denominator used in [17].

A search-free near-field source localization method with exact signal model

Most of the near-field source localization methods are developed with the approximated signal model, because the phases of the received near-field signal are highly non-linear. Nevertheless, the approximated signal model based methods suffer from model mismatch and performance degradation while the exact signal model based estimation methods usually involve parameter searching or multiple decomposition procedures. In this paper, a search-free near-field source localization method is proposed with the exact signal model. Firstly, the approximative estimates of the direction of arrival (DOA) and range are obtained by using the approximated signal model based method through parameter separation and polynomial rooting operations. Then, the approximative estimates are corrected with the exact signal model according to the exact expressions of phase difference in near-field observations. The proposed method avoids spectral searching and parameter pairing and has enhanced estimation performance. Numerical simulations are provided to demonstrate the effectiveness of the proposed method.

On the Problem of Robust Assignment of Dynamic System Poles

For a linear time-invariant single-input system with a linear inertia-free state feedback, a problem of such assignment of poles, in which bounded parametric perturbations do not bring these poles outside a certain region is considered. A generalization of the well-known solution to this problem, based on Rouche's theorem and assuming that the structure of system and the input matrices meet the consistency conditions, to linear control systems with matrices of an arbitrary form is proposed. A special parameter is introduced, on the value of which the fulfilment of the conditions of Rouche's theorem, the shape of the pole scatter region, its "volume" and the minimum attainable root measures of the dynamics performance depend. The relationship between the width of parametric uncertainty intervals and the minimum required value of the geometric mean root of the characteristic polynomial of a system with a modal controller is revealed. An example of application of the proposed technique and a computer program developed on its basis in the MATLAB language for the synthesis of a modal controller for an electromechanical system is given.

Semi-analytical solutions for eigenvalue problems of chains and periodic graphs

We show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form pn+1(z)=Qk(z)pn(z)+γpn−1(z) as n→∞, where the coefficient Qk(z) is a kth degree polynomial, and z,γ∈C. These results are then extended for approximating roots of such polynomials for any given n. Such relations for the evaluation are motivated by the large computational efforts of the iterative numerical methods in solving for eigenvalues, and their errors. We first apply this relation to the eigenvalue problems represented by tridiagonal matrices with a periodicity k in its entries, providing a more accurate method for the evaluation of the spectra of a k-periodic chain and a reduction in computing effort from O(n2) to O(n). These results are combined with the spectral rules of Kronecker products, for efficient and accurate evaluation of spectra of spatial lattices and other periodic graphs that need not be represented by a banded or a Toeplitz type matrix.

On real algebraic numbers in which the derivative of their minimal polynomial is small

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

On the structure of polynomial roots

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.

On the largest eigenvalue of a mixed graph with partial orientation

Let G be a connected graph and let T be an acyclic set of edges of G. A partial orientation σ of G with respect to T is an orientation of the edges of G except those edges of T, the resulting graph associated with which is denoted by GTσ. In this paper we prove that there exists a partial orientation σ of G with respect to T such that the largest eigenvalue of the Hermitian adjacency matrix of GTσ is at most the largest absolute value of the roots of the matching polynomial of G.

A fully nonlinear partial differential equation and its application to the σk-Yamabe problem

The Gursky-Streets equation was introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of the σ2-Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equation with uniform C1,1 estimates for 2k≤n. An important new ingredient is to show the concavity of the operator which holds for all k≤n. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform C1,1a priori estimates for all the cases n≥2k. Moreover, we establish the uniqueness of the solution to the degenerate equation for the first time.As an application, we prove that if k≥3 and M2k is conformally flat, any solution of the σk-Yamabe problem is conformal diffeomorphic to the round sphere S2k.

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

Algebraic approach to univariate polynomial derivation

Abstract Starting from n roots of a univariate polynomial of n-th degree, combining the first, the second, and up to the n-th class, new polynomials are generated. Each class of polynomials obtained in such a way is fully determined by its roots. The arithmetic means of corresponding coefficients are proportional to the values of the coefficients obtained by the derivation of the initial univariate polynomial. This gives the basis for the algebraic approach to the polynomial derivation.

Transformation of a high-order edge dislocation to optical vortices (spiral dislocations)

We theoretically show that an astigmatic transformation of an nth-order edge dislocation (a zero-intensity straight line) produces n optical elliptical vortices (spiral dislocations) with unit topological charge at the double focal distance from the cylindrical lens, located on a straight line perpendicular to the edge dislocation, at points whose coordinates are the roots of an nth-order Hermite polynomial. The orbital angular momentum of the edge dislocation is proportional to the order n.

A Diagnostic for Seasonality Based Upon Polynomial Roots of ARMA Models

Abstract Methodology for seasonality diagnostics is extremely important for statistical agencies, because such tools are necessary for making decisions whether to seasonally adjust a given series, and whether such an adjustment is adequate. This methodology must be statistical, in order to furnish quantification of Type I and II errors, and also to provide understanding about the requisite assumptions. We connect the concept of seasonality to a mathematical definition regarding the oscillatory character of the moving average (MA) representation coefficients, and define a new seasonality diagnostic based on autoregressive (AR) roots. The diagnostic is able to assess different forms of seasonality: dynamic versus stable, of arbitrary seasonal periods, for both raw data and seasonally adjusted data. An extension of the AR diagnostic to an MA diagnostic allows for the detection of over-adjustment. Joint asymptotic results are provided for the diagnostics as they are applied to multiple seasonal frequencies, allowing for a global test of seasonality. We illustrate the method through simulation studies and several empirical examples.

Impact of electric vehicles aggregators with communication delays on stability delay margins of two-area load frequency control system

This paper investigates the impact of electric vehicles (EVs) aggregator with communication time delay on stability delay margin of a two-area load frequency control (LFC) system. A frequency-domain exact method is used to calculate stability delay margins for various values of proportional-integral (PI) controller gains. The proposed method first eliminates the transcendental terms in the characteristic equation without using any approximation and then transforms the transcendental characteristic equation into a regular polynomial using a recursive approach. The key result of the elimination process is that real roots of the new polynomial correspond to imaginary roots of the transcendental characteristic equation. With the help of new polynomial, delay-dependent system stability and root tendency with respect to the time delay is determined. An analytical formula is then developed to compute delay margins in terms of system parameters. The qualitative impact of EVs aggregator on stability delay margins is thoroughly analysed and the results are verified by time domain simulations and quasi-polynomial mapping-based root finder (QPmR) algorithm.

Integral-root polynomials and chromatic uniqueness of graphs

For a graph G, a mapping f : V(G) →{1, 2, … , λ}, λ ∈ ℕ, is called a λ-colouring of G if f (u) ≠ f (v), whenever u and v are adjacent. The number of λ-colorings of a graph G is called the chromatic polynomial and denoted by P(G, λ). A polynomial P(G, λ) is called an integral-root polynomial if all its roots are integers. A graph G is an integral-root graph if P(G, λ) is an integral-root polynomial. If P(G, λ) = P(H, λ), then we say that the graphs G and H are (chromatically) equivalent (or χ-equivalent), written as G ∼ H. For any graph H, if G ∼ H implies that G is isomorphic to H, then the graph G is (chromatically) unique (or χ-unique). In this paper, our theorems cover the uniqueness problem for integral-root graphs. Altogether, summarizing them, they state the following. “If the chromatic polynomial of an integral-root graph G has exactly one root of multiplicity 2 and no more multiple root, then it is χ-unique, otherwise, G is not χ-unique”. Finally, we will be able to construct some graphs for any interval-wise integral- root polynomial.