Roots Of Polynomial Research Articles

A Method for Locating the Real Roots of the Symbolic Quintic Equation Using Quadratic Equations

AbstractA method is proposed with which the locations of the roots of the monic symbolic quintic polynomial can be determined using the roots of two resolvent quadratic polynomials: and , whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of and and on some specific relationships between them. The method is illustrated with the full analysis of one of the possible cases. Some of the roots of the symbolic quintic equation for this case have their isolation intervals determined and, as this cannot be done for all roots with the help of quadratic equations only, finite intervals containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of the quintic are identified. Knowledge of the stationary points of the quintic lifts this indeterminacy and allows finding the isolation interval of each root. Separately, using the complete root classification of the quintic, one can also lift this indeterminacy. The method also allows to see how variation of the individual coefficients of the quintic affect its roots. No root finding iterations or any numerical approximations are used and no equations of degree higher than two are solved.

Self-interaction in a cosmic dark fluid: The four-kernel rheological extension of the equations of state

We establish a new self-consistent model of coupling between the cosmic dark energy and dark matter in the framework of the rheological approach, which is based on the representation of the equations of state in terms of integral operators of the Volterra-type. We elaborate the so-called four-kernel model, in the framework of which both the dark energy and dark matter pressures are presented by two integrals containing the energy densities of the dark energy and dark matter. For the Volterra operators, the kernels of which are associated with the effects of fading memory, the corresponding isotropic homogeneous cosmological model is shown to be exactly integrable. We consider the classification of the model exact solutions, based on the analysis of roots of the characteristic polynomial associated with the key equation of the presented model. The scalars of the pressure and energy-density of the dark energy and dark matter, the Hubble function and acceleration parameter are presented explicitly as the functions of the dimensionless scale factor. The scale factor as the function of the cosmological time is found in quadratures and is described analytically, qualitatively and numerically. Asymptotic analysis allowed us to classify the models with respect to behavior typical for the Big Rip, Little Rip and Pseudo Rip (de Sitter type). Two intriguing exact cosmological solutions are discussed, which describe the super-exponential expansion and the symmetric bounce. New solutions are presented, which correspond to the quasi-periodic behavior of the state functions of the dark fluid and of the geometric characteristics of the Universe.

Classification of KPI lumps

A large family of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-defined wave patterns in the xy-plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of Painlevé II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of N objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer N.

Rationalizability of field extensions with a view towards Feynman integrals

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials.

Charged Shear-Free Fluids and Complexity in First Integrals.

The equation is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein–Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for and in terms of elementary and special functions. The explicit forms and arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral.

Paving property for real stable polynomials and strongly Rayleigh processes

One of the equivalent formulations of the Kadison–Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the “paving conjecture”. In this paper, we first extend this result to real stable polynomials. We prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the “restriction” of the polynomial to each subset has small roots. Then, we derive a probabilistic interpretation of this result. We show that there exists a partition of the underlying space of every strongly Rayleigh process into a small number of sets such that the restriction of the point process to each set has “almost independent” points. This result implies that the dependence structure of strongly Rayleigh processes is constrained—a phenomenon that is to be expected in negatively dependent measures. To prove this result, we introduce and study the notion of kernel polynomial for strongly Rayleigh processes. This notion is a natural generalization of the kernel of determinantal processes. We also derive an entropy bound for strongly Rayleigh processes in terms of the roots of the kernel polynomial which is interesting on its own.

Fast and Exact Root Parity for Continuous Collision Detection

AbstractWe introduce the first exact root parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions).Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.

Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification

AbstractThe isolation intervals of the real roots of the real symbolic monic cubic polynomial are found in terms of simple functions of the coefficients of the polynomial (such as , , , , when is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. A much‐improved complete root classification, addressing the signs (together with giving the isolation intervals) of the individual roots, is also presented. No numerical approximations or root finding techniques are used. Instead of considering the discriminant of the cubic, criterion for the existence of a single real root or three real roots is found as conditions on the coefficients of the cubic, resulting from the roots of the auxiliary quadratic equations. It is also shown that, if a cubic equation has three real roots, then these lie in an interval such that , independent of . A detailed algorithm for applying the method for isolation of the roots of the cubic is also given and it is illustrated through examples, including the full mathematical analysis of the cubic equation associated with the Rayleigh elastic waves and finding the isolation intervals of its real roots.

Linear instability of viscoelastic interfacial Hele-Shaw flows: A Newtonian fluid displacing an UCM fluid

We theoretically study the linear stability of the Saffman–Taylor problem where a viscous Newtonian fluid (with viscosity η l ) displaces an Upper Convected Maxwell (UCM) fluid (with viscosity η r ) in a rectilinear Hele-Shaw cell. The dispersion relation is given by the roots of a cubic polynomial with coefficients depending on wavenumber along with several dimensionless groups as parameters. Using Routh–Hurwitz stability criterion, we show that the viscosity ratio η r / η l still plays a decisive role in determining stability (stable if η r / η l ≤ 1 ). If η r / η l > 1 , the flow is more unstable than an identical Newtonian–Newtonian setup and the most unstable wavenumber is larger. Increasing Deborah number, capillary number or flow speed worsens the instability. Elasticity has a variety of effects and can give rise up to three types of singular behaviors: (i) there exists infinitely many distinct wavenumbers at which the velocity becomes singular, (ii) stress becomes singular when the wavenumber exceeds a certain value; and (iii) a resonance phenomenon occurs when η r / η l is large, where the growth rate increases very rapidly near certain wavenumber and eventually becomes singular when the displacing fluid is inviscid. • Explicit formulas for the dispersion relation. • Singular behaviors in the velocity, stress, and growth rate. • Viscosity contrast determines long wave stability. • Destabilizing effect if increasing flow speed, Deborah number or Capillary number. • More unstable than Newtonian case with shorter most unstable wave.

The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation

The isolation intervals of the real roots of the symbolic monic cubic polynomial x^3 + a x^2 + b x + c are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle—the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.

Reliable event detection for Taylor methods in astrodynamics

ABSTRACT We present a novel approach for the detection of events in systems of ordinary differential equations. The new method combines the unique features of Taylor integrators with state-of-the-art polynomial root finding techniques to yield a novel algorithm, ensuring strong event detection guarantees at a modest computational overhead. Detailed tests and benchmarks focused on problems in astrodynamics and celestial mechanics (such as collisional N-body systems, spacecraft dynamics around irregular bodies accounting for eclipses, computation of Poincaré sections, etc.) show how our approach is superior in both performance and detection accuracy to strategies commonly employed in modern numerical integration works. The new algorithm is available in our open source Taylor integration package heyoka.

On the dynamics of the roots of polynomials under differentiation

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any initial data in the Sobolev space Hs(S) with s>3/2. In this paper, we consider the Cauchy problem in the critical space H1/2(S). Two interesting new features, at this level of regularity, are that the equation can be written in the form∂tu+V∂xu+γΛu=0, where V is not bounded and γ is not bounded from below. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any initial data in H1/2(S). Our proof relies on sharp commutators estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.

Dynamics of KPI lumps

A family of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the xy-plane are altered after collision. Thus they differ from the standard multi-peaked KPI simple n-lump solutions whose peak heights as well as peak trajectories remain unchanged after interaction. The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomena is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long time asymptotics of the solutions that the peak trajectories separate as as |t| → ∞, and all the peak heights approach the same constant value corresponding to that of the simple one-lump solution. Consequently, a multi-peaked n-lump solution evolves to a superposition of n one-lump solutions asymptotically as |t| → ∞.

Walsh–Hadamard transforms of generalized [formula omitted]-ary functions and [formula omitted]-finite sequences

In this article we show that Walsh–Hadamard transformations of generalized p-ary functions whose components are symmetric, rotation symmetric or a combination or concatenation of them are C-finite sequences. This result generalized many of the known results for regular p-ary functions. We also present a study of the roots of the characteristic polynomials related to these sequences and show that properties like balancedness and being bent are not shared by the underlying p-ary functions.

Synthesis Based on Linearization of Vector Speed Control of an Induction Motor without a Speed Sensor

The aim of the work was parametric synthesis of vector sensorless (i. e. without speed sensor) control of an electric drive with an induction motor. The structure of the system is based on the application of an adaptive model for estimating the rotor flow coupling vector and velocity. The speed is estimated by the mismatch of the real stator current and the current value calculated in the model. Stability is guaranteed in this well-known structure, obtained on the basis of Lyapunov functions, but it remains problematic to calculate the parameters of regulators and an adaptive model to ensure high-quality dynamics of the system. For a vector control system of an induction electric motor without a speed sensor with an adaptive model, a linearized structure in a synchronously rotating coordinate system was proposed. This makes it possible to calculate control parameters using the modal control method to ensure quality indicators in each of the closed circuits of the system. Such parametric synthesis is based on the assumption that the flow coupling of the rotor is maintained constant, and therefore the mutual influence of the flow coupling and torque control channels can be neglected. The calculation of the parameters of control (regulators and the adaptation channel) is based on the method of assigning the roots of characteristic contour polynomials in such a way that each internal contour has a higher speed than the external one with respect to it. The method is approximate, but it makes it possible to take into account the main cause-and-effect relationships in dynamics and obtain simple calculation expressions. The simulation of the system was carried out using a simulation model that takes into account the digital software-algorithmic method for generating a microcontroller control signal, as well as electromagnetic processes under conditions of pulse-width modulation in an electric energy converter and an electric motor, the use of the values of the rotor flow coupling vector estimated by the model in coordinate transformations of the system, the formation of a spatial vector of the converter voltage. The analysis of the synthesized speed control system by the simulation method has confirmed the effectiveness of the proposed method of parametric synthesis and the acceptable accuracy of speed estimation.

Estimation of Transport-Category Jet Airplane Maximum Range and Airspeed in the Presence of Transonic Wave Drag

One of the most difficult steps in estimating the cruise performance characteristics of high-subsonic transport-category turbofan-powered airplanes is the estimation of the transonic wave drag. Modern jet airplanes cruise most efficiently in the vicinity of the drag-divergence or drag-rise Mach numbers. In the initial design phase and later when the preliminary wind-tunnel and/or CFD computations and drag polars are known with increased accuracy, a method of estimating cruise performance is needed. In this study, a new semi-empirical transonic wave drag model using modified Lock’s equation was developed. For maximum range cruise estimations, an optimization criterion based on maximizing specific air range was used. The resulting nonlinear equations are of 12th- and 13th-order. Numerical Newton–Raphson nonlinear solvers were used to find real positive roots of such polynomials. The NR method was first tested for accuracy and convergence using known analytical solutions. A methodology for an initial guess was developed starting with the maximum-range cruise Mach without the wave-drag included. This guess resulted in fast quadratic convergence in all computations. Other novel features of this article include a new semi-empirical fuel-flow law, which was also extensively tested. Additionally, a semi-empirical turbofan thrust model usable for a wide range of bypass ratios and the entire flight envelope was developed. Such physics-based semi-empirical model can be used for a wide range of turbofans. The algorithm can be utilized to identify most beneficial input parameter values and combinations for the cruise flight phase. The model represents a powerful tool to estimate important cruise performance airspeeds located in the transonic regime. An intended application is in the conceptual development stages for early design optimizations of future airplanes. It is possible with additional effort to extend existing model capabilities to deal with supersonic transports optimal cruise parameters.

Hyperbolic Quadrature Method of Moments for the One-Dimensional Kinetic Equation

A tractable solution is proposed to a classical problem in kinetic theory, namely, given any set of realizable velocity moments up to order $2n$, a closure for the moment of order $2n+1$ is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order ($n \le 2$). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials $Q_n$ that are uniquely defined by the velocity moments up to order $2n-1$. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, $n$ double roots of the characteristic polynomial $P_{2n+1}$ of the Jacobian matrix of the system are the roots of $Q_n$, while in the interior, $P_{2n+1}$ and $Q_n$ share $n$ roots. The remaining $n+1$ roots of $P_{2n+1}$ bound and separate the roots of $Q_n$. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order $2n+1$ from the moments up to order $2n$ is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing $n$.

A note on non-monogenity of number fields arised from sextic trinomials

Let K = Q (θ) be an algebraic number field with θ a root of an irreducible polynomial f(x) = x 6 + axm + b belonging to ℤ[x] and 1 ≤ m ≤ 5. Let OK be the ring of algebraic integers of K. We say that K is monogenic if there exists some α ∈ OK such that OK = Z[α]. In this paper, we give some explicit conditions on a, b for which K is non-monogenic. We illustrate our results through examples.

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.

A Laplace transform approach to direct and inverse problems for multi-compartment models

Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.