Polynomial Decomposition Research Articles

Decomposition of polynomials using the shift-composition operation

Abstract V. I. Solodovnikov had employed the shift-composition operation to investigate homomorphisms of shift registers into linear automata; in his papers, conditions for the absence of nontrivial inner homomorphisms of shift registers were derived. An essential role was played by the condition of linearity of the left component of the shift-composition operation in the corresponding polynomial decomposition. In this paper we consider the case where the left component is a function belonging to a wider class, which includes the class of linear functions.

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

A Quaternionic Bernstein Theorem

We prove a four-dimensional version of a Bernstein’s theorem, with complex polynomials being replaced by quaternionic polynomials. Moreover, using an Almansi-type decomposition of polynomials, we formulate the quaternionic Bernstein’s inequality in terms of four-dimensional zonal harmonics and Gegenbauer polynomials.

Heat coefficients for magnetic Laplacians on the complex projective space P n(ℂ)

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν .

Quadratic Decomposition of Bivariate Orthogonal Polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Bäcklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

Power-product matrix: nonsingularity, sparsity and determinant

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained.

On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. If G is a connected reductive group defined over a non-archimedean local field 𝔉, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of G(𝔉) decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of G. We classify the non-obtuse parabolics. This then shows that our decomposition theorem covers all the classical cases due to Andrianov and Gritsenko. We also obtain new cases when the relative root system of G contains factors of types E 6 or E 7 .

Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients

Assessing nonnegativity of multivariate polynomials over the reals, through the computation of certificates of nonnegativity, is a topical issue in polynomial optimization. This is usually tackled through the computation of sum of squares decompositions which rely on efficient numerical solvers for semidefinite programming. This method faces two difficulties. The first one is that the certificates obtained this way are approximate and then nonexact. The second one is due to the fact that not all nonnegative polynomials are sums of squares. In this paper, we build on previous works by Parrilo and Nie, Demmel, and Sturmfels who introduced certificates of nonnegativity modulo gradient ideals. We prove that, actually, such certificates can be obtained exactly over the rationals if the polynomial under consideration has rational coefficients, and we provide exact algorithms to compute them. We analyze the bit complexity of these algorithms and deduce bitsize bounds of such certificates.

Adaptive Fuzzy Sliding Exact Tracking Control Based on High-Order Log-Type Time-Varying BLFs for High-Order Nonlinear Systems

This article investigates the adaptive fuzzy sliding exact tracking control problem of a class of high-order nonlinear systems with full-state constraints and mismatched external disturbances. To achieve exact tracking control, the upper bound of a composite polynomial is estimated, which is composed of approximation error, mismatched disturbance, and virtual control law and its upper bound estimated information is employed in the controller design. In this way, once the tracking error deviates from the origin, the corresponding adaptive fuzzy sliding exact tracking control mechanism will be activated to force it to slide along the origin, and then exact tracking control is achieved. Meanwhile, high-order log-type time-varying barrier Lyapunov functions are introduced to ensure that the full-state constraint is not violated. A 2-D high-order nonlinear system and an underactuated nonlinear benchmark system with weak coupling are considered to illustrate the characteristics of the control mechanism in simulation.

Inequalities for a class of composite polynomials

In this paper, we establish some inequalities for a more general class of polynomials P(S(z)), where P(z) = ∑ ajzj and S(z) = ∑ bjzj of degree n and m respectively. These results not only generalize the results due to Dubinin [ J. Math. Sci., 143 (2007), 3069-3076] but also refine a result due to Shah and Liman [ NFAA, 9(2004), 223-232].

Minimax Approximation of Sign Function by Composite Polynomial for Homomorphic Comparison

The comparison operation for two numbers is one of the most frequently used operations in several applications, including deep learning. As such, lots of research has been conducted with the goal of efficiently evaluating the comparison operation in homomorphic encryption schemes. Recently, Cheon <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (Asiacrypt 2020) proposed new comparison methods that approximated the sign function on homomorphically encrypted data using composite polynomials and proved that these methods had optimal asymptotic complexity. In this article, we propose a practically optimal method that approximates the sign function using compositions of minimax approximation polynomials. We prove that this approximation method is optimal with respect to depth consumption and the number of non-scalar multiplications. In addition, we propose a polynomial-time algorithm that determines the optimal composition of minimax approximation polynomials for the proposed homomorphic comparison operation using dynamic programming. The numerical analysis demonstrates that when minimizing runtime, the proposed comparison operation reduces the runtime by approximately 45 percent on average when compared to the previous algorithm. Likewise, when minimizing depth consumption, the proposed algorithm reduces the runtime by approximately 41 percent on average. In addition, when high precision in the comparison operation is required, the previous algorithm does not achieve 128-bit security, while the proposed algorithm does due to its small depth consumption.

Об управляемости и наблюдаемости многоканальных САУ при синтезе модальным методом с использованием полиномиального матричного разложения объекта и регулятора

When considering the issue of synthesis of an automatic control system by a modular method using a polynomial decomposition of the transfer functions of an object and a controller, synthesis algorithms for fully controlled systems are proposed. However, the question arises about the possibilities of using this algorithm if this condition is not met. The consideration of this is-sue turned out to be especially relevant for multichannel models of objects with a non-square transfer function (having an unequal number of input and output channels). It is shown that for some fundamental terms of the theory of automatic control, such as controllability, reachability, observability, stability and some others, there are special definitions of them in the case of considering this type of objects. The term non‒square object is proposed for use, which is used mainly in foreign literature. Some restrictions on the modal synthesis of regulators by a method using a polynomial matrix separation of the object and the regulator are considered. Examples of internally and asymptotically unstable systems are given. A hypothesis is put forward about the stability of the controlled system. An example of a multichannel system "inverted pendulum on a cart" is considered, which is an object with a non-square matrix transfer function (in this example, the number of input actions is less than the number of output parameters). Using the static characteristics of this object, it is demonstrated that not always controlled systems can be stabilized in a given position. For example, in the case of setting the desired angle of an inverted pendulum other than zero, it is impossible to hold the position of the cart in a given coordinate. At the same time, if you set the angle at the equilibrium point as the desired angle of the inverted pendulum, then stabilization of the cart at a given coordinate becomes possible.

Wavefront error based tilt-to-length noise analysis for the LISA transmitted beam

The laser interferometer space antenna (LISA) will open the signal-rich 100 μHz to 1 Hz gravitational wave window. LISA is expected to be limited by acceleration noise in the low frequency range and noise associated with the optical measurement system above a few mHz. Of the latter, apparent length changes due to spacecraft (SC) angular jitter are among the most critical contributors. One of the coupling mechanisms is via wavefront error in the transmitted beam. Utilizing a Zernike polynomial decomposition of such wavefront error, we introduce and explore the validity of extremely fast best fit polynomial expansion based noise recreation tools that provide a clear picture for which transmit beam perturbations couple most strongly with SC jitter into LISA noise.

Semi-analytic Fresnel diffraction calculation with polynomial decomposition.

The numerical method based on the fast Fourier transform (FFT) is generally applied to calculate the Fresnel diffraction field, which would suffer from sampling constraints. To break this limit, in this Letter, the semi-analytic Fresnel diffraction calculation method is proposed based on polynomial decomposition. The diffraction field is computed by using properly analytic Fresnel diffraction basis functions (FDBFs) according to the application requirements. Analytic FDBF is calculated based on Legendre or Chebyshev polynomials by using the object-domain frequency division multiplexing method. The proposed method offers arbitrary sampling, high-flexibility, and high-accuracy diffraction calculation in the full Fresnel region. The computational efficiency and accuracy of the proposed method are compared with FFT-based methods. It has potential application in light field analysis, wavefront sensing, and image processing.

On decompositions of permutation polynomials into quadratic and cubic power permutations

On decompositions of permutation polynomials into quadratic and cubic power permutations

On decompositions of binary recurrent polynomials

Polynomial decomposition expresses a polynomial f as the functional composition f=gcirc h of lower degree polynomials g and h, and has various applications. In this paper, we will show that for a minimal, non-degenerate, simple, binary, linearly recurrent sequence (G_n(x))_{n=0}^infty of complex polynomials whose coefficients in the Binet form are constants, if G_n(x)=g(h(x)), then apart from some exceptional situations that have to be taken into account, the degree of g is bounded by a constant independent of n. We will build on a general but conditional result of this type that already exists in the literature. We will then present one Diophantine application of the main result.

Interval Uncertainty Quantification for the Dynamics of Multibody Systems Combing Bivariate Chebyshev Polynomials with Local Mean Decomposition

Interval quantification for multibody systems can provide an accurate dynamic prediction and a robust reliability design. In order to achieve a robust numerical model, multiple interval uncertain parameters should be considered in the uncertainty propagation of multibody systems. The response bounds obtained by the bivariate Chebyshev method (BCM) present an intensive deterioration with the increase of time history in the interval dynamic analysis. To circumvent this problem, a novel method that combines the bivariate Chebyshev polynomial and local mean decomposition (BC-LMD) is proposed in this paper. First, the multicomponent response of the system was decomposed into the sum of several mono-component responses and a residual response, and the corresponding amplitude and phase of the mono-component were obtained. Then, the bivariate function decomposition was performed on the multi-dimensional amplitude, phase, and residual to transform a high-dimensional problem into several one-dimensional and two-dimensional problems. Subsequently, a low order Chebyshev polynomial can be used to construct surrogate models for the multi-dimensional amplitude, phase, and residual responses. Then, the entire coupling surrogate model of the system can be established, and the response bounds of the system can be enveloped. Illustrative examples of a slider-crank mechanism and a double pendulum are presented to demonstrate the effectiveness of the proposed method. The numerical results indicate that, compared to the BCM, BC-LMD can present a tight envelope in the long time-dependent dynamic analysis under multiple interval parameters.

Analyzing the dual space of the saturated ideal of a regular set and the local multiplicities of its zeros

In this paper, a method is proposed to analyze the structure of dual space of the saturated ideal generated by a regular set and the local multiplicities of its zeros. In detail, we generalize the squarefree decomposition of univariate polynomials to the so‐called pseudo squarefree decomposition of multivariate polynomials and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algorithm, the multiplicities of zeros could be directly read out, and the real solution isolation with multiplicity can also be easily produced.

An Elementary Proof of Takagi’s Theorem on the Differential Composition of Polynomials

I give a short and completely elementary proof of Takagi’s 1921 theorem on the zeros of a composite polynomial .

Validation of solid mechanics models using modern computation techniques of Zernike moments

Numerical simulation plays an vital role in mechanical engineering, as it allows the prediction of the mechanical behavior of a component, contributing to the improvement of its performance and the optimization of the design. However, before using the numerical model’s valuable information, its reliability must be verified in a process called validation. Validation is usually carried out experimentally, measuring the mechanical behavior of a mechanical part or component. If there is a good agreement between the model’s prediction and the experimental measurements, the numerical simulation and its predictions are validated. Traditionally, the validation technique consists of identifying the critical areas of the component and checking the agreement between the simulation and the measurements by means of strain gauges located in these areas. More recently, optical techniques for the deformations measurement have gained much interest due to their ability to measure the entire extension of a component captured by the image. This property, called full-field measurement, allows the validation of the whole numerically simulated sample, improving the reliability and precision of the validation. In this procedure, the comparison between the simulated and experimental data requires an image compression process, typically using descriptors based on Zernike moments or any polynomial decomposition. This article presents an improved validation process based on the efficient and accurate calculation of Zernike moments. Concerning the currently used procedures, this new procedure makes it possible to significantly increase the order of image decomposition, the size (i.e., the resolution) of the analyzed region of interest (ROI), and reduce the processing time. All this results in an improvement in validation reliability and the possibility of validating strains/displacements with arbitrary shapes, even with discontinuities.