Roots Of Polynomial Research Articles

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

In this note, we develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's “polygon rule”.

Min-max elementwise backward error for roots of polynomials and a corresponding backward stable root finder

A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials p(z). Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of p(z) that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of p(z). It uses a companion linearization C(z)=A−zB of p(z) to which we added an extra zero leading coefficient, and an appropriate two-sided diagonal scaling that balances A and makes B graded in particular when there is variation in the magnitude of the coefficients of p(z). An implementation of the QZ algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of p(z). Under the assumption that this implementation of the QZ algorithm exhibits a graded backward error when B is graded, we prove that our new algorithm is min-max elementwise backward stable. Several numerical experiments show the superior performance of the new algorithm compared with the MATLAB roots function. Extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers, as illustrated by many numerical tests.

Monostatic MIMO radar with nested L-shaped array: Configuration design, DOF and DOA estimation

In this paper, we consider the problem of two-dimensional (2-D) direction of arrival (DOA) estimation for L-shaped monostatic MIMO radar with two-level nested linear array (NLA). To obtain more degrees of freedom (DOFs) than traditional nested L-shaped (NLs) MIMO radar, we propose the extensional NLs (ENLs) MIMO radar by extending all of the sensor positions in transmitter array with an identical magnification compared with NLs MIMO radar. The ENLs MIMO radar can offer O(M1M2N1N2) DOFs with only O(M1+M2+N1+N2) sensors, where M1(N1) and M2(N2) respectively represents the numbers of the first-level and second-level sensors of NLA in transmitter (receiver) array. Furthermore, to avoid the eigenvalue decomposition, spatial spectral search and polynomial root finding technique in spatial smoothing multiple signals classification (SS-MUSIC) and SS-ROOT-MUSIC, and to improve the estimation accuracy simultaneously, we propose a search-free iterative Taylor expansion (SF-ITE) algorithm to perform DOA estimation, which employs the Discrete Fourier transform (DFT) method to get the coarse estimation, and then utilizes the iterative Taylor expansion technique to obtain fine estimation. SF-ITE can get more precise estimation by performing iteration only once than both SS-MUSIC and SS-ROOT-MUSIC. In addition, we derive an angle-pairing procedure to help to eventually resolve more targets than the total number of sensors. Finally, simulation results are provided to validate the superiority of the ENLs MIMO radar and SF-ITE algorithm.

Distribution of Complex Algebraic Numbers on the Unit Circle

For $-\pi\leq\beta_1<\beta_2\leq\pi$ denote by $\Phi_{\beta_1,\beta_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[\beta_1,\beta_2]$ of degree $2m$ and with elliptic height at most $Q$. We show that \[ \Phi_{\beta_1,\beta_2}(Q)=Q^{m+1}\int\limits_{\beta_1}^{\beta_2}{p(t)}\,{\rm d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to a constant factor with the density of the roots of some random trigonometric polynomial. This density is calculated explicitly using the Edelman--Kostlan formula.

104.33 Bounds for roots of polynomials with increasing coefficients

104.33 Bounds for roots of polynomials with increasing coefficients

2D-DOA Estimation of Strictly Noncircular Sources Utilizing Connection-Matrix for L-Shaped Array

In wireless communication, the signals with noncircularity are widely applied. In this letter, we consider the problem of two-dimensional direction of arrival estimation with L-shaped array for strictly noncircular sources. To obtain more precise estimation with lowest possible computational complexity, we construct two connection-matrices to establish the relation between the data corresponding to the two orthogonal uniform linear arrays in L-shaped array so that we can still exploit all information of the extended received signal after dimensionality reduction. On this basis, the rank-reduction technique and one-dimensional polynomial root finding technique are able to be utilized only twice to obtain unpaired estimation results. Finally, a concise angle pairing is performed via the third connection-matrix. Compared with existing algorithms, the proposed algorithm can obtain more precise estimation and better resolution. Simulation results are provided to validate the superiority of the proposed algorithm.

A fourth-order cumulant orthonormal propagator rooting method based on Toeplitz approximation

A novel Toeplitz fourth-order cumulant (FOC) orthonormal propagator rooting method (TFOC ‐ OPRM) of direction-of-arrival (DOA) estimation for uniform linear array (ULA) is proposed in this paper. Specifically, the modified (i.e., reduced-dimension) FOC (MFOC) matrix is achieved at first via removing the redundant information encompassed in the primary FOC matrix; then, the TFOC matrix which possesses Toeplitz structure can be recovered by utilizing the Toeplitz approximation method. To reduce the computational complexity, an effective method based on the polynomial rooting technology is adopted. Finally, the DOAs of incident signals can be estimated by exploiting orthonormal propagator rooting method. The theoretical analysis coupled with simulation results show that the proposed resultant algorithm can reduce the computational complexity significantly, as well as improve the estimation performance in both spatially white noise environment and spatially color noise environment.

Coulson-type integral formulas for the sum of powers of absolute values of roots of polynomials

Coulson integral formula was initiated in the research of energy of graphs in 1940 by Coulson, which provides an efficient way to calculate the energy of graphs without knowing any eigenvalues of graphs. Recently, Qiao et al. generalized the concept of energy from graphs to polynomials, which is referred to as the general energy of polynomials. For a polynomial ϕ(z) and a nonzero real number α, the general energy of polynomial ϕ(z), denoted by Eα(ϕ), is defined as the sum of αth powers of absolute values of all (nonzero) roots of ϕ(z), i.e., Eα(ϕ)=∑zk≠0|zk|α, where zk represents the nonzero roots of ϕ(z). In a series of three publications by Qiao et al., they gave some beautiful Coulson-type integral formulas for Eα(ϕ) when the roots of ϕ(z) are all nonnegative numbers for any rational number α and any irrational number α with 0<|α|<1, and when the roots of ϕ(z) are all real numbers for any rational number α and any irrational number α with 0<|α|<2. For the remaining α, their integral formulas are invalid. In this paper, we set up a research procedure covering all the nonzero real numbers α. That is to say, it not only combines all the Coulson-type integral formulas established by Qiao et al., but also involves in all the remaining unsolved α. As an extension and generalization of known integral formulas, the Coulson-type integral formulas for Eα(ϕ), when the roots of ϕ(z) are all pure imaginary numbers, are also presented, for any nonzero real number α. Moreover, we also give an alternative form of integral formulas expressed by the coefficients of polynomials.

Stress formalism for elastic waves in anisotropic solids

Traditionally, the propagation of bulk elastic waves in anisotropic crystals or solids is governed by the displacement form of the elastodynamic wave equation. Phase velocities follow after transforming the wave equation into an eigenvalue problem, which gives the Christoffel equations, and solving for the roots of the characteristic polynomial. Alternatively, only a few researchers have considered a pure stress wave equation of motion to describe the propagation of waves in elastic solids. However, to our knowledge, the dual characteristic polynomial and its solutions based on a pure stress formalism has not been established previously. This presentation will outline the needed steps to reach and solve the dual characteristic polynomial. Phase velocity solutions follow from the formation of new principal invariants, which are functions of the elastic compliance constants and propagation directions. The dual formalism is proven consistent to the traditional displacement formalism through the use of sixth-rank Levi-Civitia identities. Lastly, possible extensions of this work will be highlighted.

Quadratic Extensions in ACL2

Given a field K, a quadratic extension field L is an extension of K that can be generated from K by adding a root of a quadratic polynomial with coefficients in K. This paper shows how ACL2(r) can be used to reason about chains of quadratic extension fields Q = K_0, K_1, K_2, ..., where each K_i+1 is a quadratic extension field of K_i. Moreover, we show that some specific numbers, such as the cube root of 2 and the cosine of pi/9, cannot belong to any of the K_i, simply because of the structure of quadratic extension fields. In particular, this is used to show that the cube root of 2 and cosine of pi/9 are not rational.

Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

English

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation A ⊗ x = λ ⊗ x for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if λ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial.

An efficient nonlinear polynomial color characterization method based on interrelations of color spaces

AbstractA well‐known color characterization method is to take an image of a color chart and then to find the mapping matrix from the digital RGBs to the corresponding known CIEXYZs. However, the prediction errors are generally large in CIELAB color space because of the nonlinear transformation from CIEXYZs to CIELABs. In this article, we propose an efficient and simple nonlinear method for the color characterization of input devices. The approach for deriving a colorimetric mapping between digital RGB signals and CIELAB tristimulus values uses the polynomial modeling by considering the interrelations among the standard CIE color spaces. Furthermore, to improve the accuracy of solution, we take the polynomial root terms extension. Our algorithm is simple to implement because only a least‐squares mapping should be solved. Various computational results are given to demonstrate the efficiency and capability of the proposed method.

The radial part of a class of Sobolev polynomials on the unit ball

In this work, a Sobolev inner product on the unit ball of $\mathbb {R}^{d}$ involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner product. The properties of this family constitute our main subject of study. In particular, we deduce algebraic properties, connection formulas, and some asymptotic properties. Finally, we show some numerical tests to illustrate the behavior of the roots of these univariate non-standard orthogonal polynomials.

Note on some representations of general solutions to homogeneous linear difference equations

It is known that every solution to the second-order difference equation x_{n}=x_{n-1}+x_{n-2}=0, nge 2, can be written in the following form x_{n}=x_{0}f_{n-1}+x_{1}f_{n}, where f_{n} is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

Free fermionic and parafermionic quantum spin chains with multispin interactions

We introduce a family of $Z(N)$ multispin quantum chains with a free fermionic ($N=2$) or free parafermionic ($N&gt;2$) eigenspectrum. The models have $(p+1)$ interacting spins ($p=1,2,\ensuremath{\cdots}$), which are Hermitian in the $Z(2)$ (Ising) case and non-Hermitian for $N&gt;2$. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order $(p+1)$. In the critical limit we identify these polynomials with certain hypergeometric polynomials ${}_{p+1}{F}_{p}$. Also in the critical regime, we calculate the ground-state energy in the bulk limit and verify that they are given in terms of the Lauricella hypergeometric series. The models with special couplings are self-dual and at the self-dual point show a critical behavior with a dynamical critical exponent ${z}_{c}=\frac{p+1}{N}$.

Computing one billion roots using the tangent Graeffe method

Let p be a prime of the form p = σ2 k + 1 with σ small and let F p denote the finite field with p elements. Let P ( z ) be a polynomial of degree d in F p [ z ] with d distinct roots in F p . For p =5 · 2 55 + 1 we can compute the roots of such polynomials of degree 10 9 . We believe we are the first to factor such polynomials of size one billion. We used a multi-core computer with two 10 core Intel Xeon E5 2680 v2 CPUs and 128 gigabytes of RAM. The factorization takes just under 4,000 seconds on 10 cores and uses 121 gigabytes of RAM. We used the tangent Graeffe root finding algorithm from [27, 19] which is a factor of O (log d ) faster than the Cantor-Zassenhaus algorithm. We implemented the tangent Graeffe algorithm in C using our own library of 64 bit integer FFT based in-place polynomial algorithms then parallelized the FFT and main steps using Cilk C. In this article we discuss the steps of the tangent Graeffe algorithm, the sub-algorithms that we used, how we parallelized them, and how we organized the memory so we could factor a polynomial of degree 10 9 . We give both a theoretical and practical comparison of the tangent Graeffe algorithm with the Cantor-Zassenhaus algorithm for root finding. We improve the complexity of the tangent Graeffe algorithm by a factor of 2. We present a new in-place product tree multiplication algorithm that is fully parallelizable. We present some timings comparing our software with Magma's polynomial factorization command. Polynomial root finding over smooth finite fields is a key ingredient for algorithms for sparse polynomial interpolation that are based on geometric sequences. This application was also one of our main motivations for the present work.

Efficient Private Set Intersection Using Point-Value Polynomial Representation

Private set intersection (PSI) allows participants to securely compute the intersection of their inputs, which has a wide range of applications such as privacy-preserving contact tracing of COVID-19. Most existing PSI protocols were based on asymmetric/symmetric cryptosystem. Therefore, keys-related operations would burden these systems. In this paper, we transform the problem of the intersection of sets into the problem of finding roots of polynomials by using point-value polynomial representation, blind polynomials’ point-value pairs for secure transportation and computation with the pseudorandom function, and then propose an efficient PSI protocol without any cryptosystem. We optimize the protocol based on the permutation-based hash technique which divides a set into multisubsets to reduce the degree of the polynomial. The following advantages can be seen from the experimental result and theoretical analysis: (1) there is no cryptosystem for data hiding or encrypting and, thus, our design provides a lightweight system; (2) with set elements less than 212, our protocol is highly efficient compared to the related protocols; and (3) a detailed formal proof is given in the semihonest model.

Polynomization of the Bessenrodt\u2013Ono Inequality

In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials P_n(x). We prove for all real numbers x >2 and a,b in mathbb {N} with a+b >2 the inequality: Pa(x)·Pb(x)>Pa+b(x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} P_a(x) \\, \\cdot \\, P_b(x) > P_{a+b}(x). \\end{aligned}$$\\end{document}We show that P_n(x) < P_{n+1}(x) for x ge 1, which generalizes p(n) < p(n+1), where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: P_2(-3+ sqrt{10}) = P_{3}(-3 + sqrt{10}).

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.