Affine Polynomial Research Articles

The hybrid method for finding the roots of a polynomial over finite fields based on affine expansion

This paper proposes a hybrid method to find the roots of a polynomial in a finite field based on combining the polynomial decomposition into the sum of multiples of the affine polynomial with the analytical method to find the roots of low-order polynomials. The proposed method allows a significant reduction in complexity and is applicable in the design of low-latency BCH and Reed-Solomon decoders.

Robust-PID controller design for magnetic levitation system using parameter space approach

Recently, implementation of air magnetic suspension systems has become more challenging due to the growing demand for lightweight technologies, comfortable cabins, vehicle safety stability, and pollution control. Magnetic levitation, or Maglev, is the method of propelling a target into the air by adjusting different magnetic forces. The absence of contact and the avoidance of wear and friction phenomena are key considerations in the applications of magnetic levitation technology. Due to the high nonlinearity in the modelling process of such kind of systems, the stabilizing of magnetic levitation has been considered as a challenging task for many researchers in control engineering sector. The computation of all stabilizing PID gains controllers for the magnetic levitation benchmark ED-4810 (Maglev) is demonstrated in this paper using two different scenarios. In the first one, the tuning parameters of the classical PID controller (KP, KI, and KD) are assumed to be the uncertain parameters. The second scenario demonstrates the uncertainty in the transfer function of the system by using the resistance and the inductance as uncertain parameters. The characteristic polynomial of the linearized uncertain model is shown to be an unstable affine polynomial using the Zero Exclusion Theorem and the singular frequencies technique. The parameter space approach is used to illustrate the values of all PID parameters in order to achieve robust stability in the two scenarios. The effectiveness of the presented graphical technique has been verified through MATLAB simulation to obtain robust stability for magnetic levitation systems.

Low latency BCH decoder using the affine polynomial over the finite field

The paper proposes a low latency BCH decoder with low complexity using parallel computation and simplifying locating errors by finding the roots of the affine polynomial over finite fields. The proposed design can be implemented on low-cost hardware platforms while applicable in very low latency information systems.

Construction of Permutation Polynomials Using Additive and Multiplicative Characters

Permutation is a natural phenomenon useful for understanding and explaining the structural and functional behavior of objects or concepts. The mathematical formulation of permutation behavior can be readily achieved by permutation polynomials. Permutation polynomials are constructed by suitably modifying linearized polynomials and associated affine polynomials with the help of additive characters, multiplicative characters, and special types of Trace functions for the polynomials in one and more than one indeterminates. The permutation properties of the obtained polynomials are verified using the AGW criterion.

A-Posteriori-Steered $p$-Robust Multigrid with Optimal Step-Sizes and Adaptive Number of Smoothing Steps

We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p ≥ 1$. Our solver employs zero pre- and one post-smoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of Schoberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of Xu et al. [Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659]. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.

MODIFIED ALGORITHM FOR SEARCHING THE ROOTS OF THE ERROR LOCATORS POLYNOMINAL WHILE DECODING BCH CODES

Context. In telecommunications and information systems with an increased noise component the noise-resistant cyclic BCH and Reed-Solomon codes are used. The adjustment and correcting errors in a message require some effective decoding methods. One of the stages in the procedure of decoding RS and BCH codes to determine the position of distortions is the search for the roots of the error locator polynomial. The calculation of polynomial roots, especially for codes with significant correction capacity is a laborious task requiring high computational complexity. That is why the improvement of BCH and RS codes decoding methods providing to reduce the computational complexity is an urgent task.Objective. The investigation and synthesis of the accelerated roots search algorithm of the error locator polynomial presented as an affine polynomial with coefficients in the finite fields, which allows accelerating the process of BCH and RS code decoding.Method. The classical roots search method based on the Chan’s algorithm is performed using the arithmetic of the Galois finite fields and the laborious calculation, in this case depends on the number of addition and multiplication operations. For linearized polynomials, the roots search procedure based on binary arithmetic is performed taking into account the values obtained at the previous stages of the calculation, which provides the minimum number of arithmetic operations.Results. An accelerated algorithm for calculating the values of the error locator polynomial at all points of the GF(2m) finite field for linearized polynomials based on the Berlekamp-Massey method has been developed. The algorithm contains a minimum number of addition operations, due to the use at each stage of the calculations the values obtained at the previous step, as well as the addition in the finite field GF(2). A modified roots search method for affine polynomials over the finite fields has been proposed to determine error positions in the code word while decoding the cyclic BCH and RS codes.Conclusions. The scientific newness of the work is to improve the algorithm of calculating the roots of the error locator polynomial, which coefficients belong to the elements of the finite field. At the same time it simplifies the procedure for cyclic BCH and RS codes decoding, due to reducing the computational complexity of one of the decoding stages, especially finding the error positions using the modified Berlekamp-Massey algorithm. These facts are confirmed by the simulation program results of the roots search of the error locator polynomial algorithm. It is shown, that the application of the accelerated method permits to reach a gain on speed of 1.5 times.

Simultaneous stabilization of polynomial nonlinear systems via density functions

This paper is concerned with simultaneous stabilization of a class of polynomial nonlinear systems with almost stability theory. The theory uses dual Lyapunov or density function criterion which has remarkable advantages over Lyapunov-based simultaneous stabilization. These advantages rely in convexity property of density functions. This property results in controller synthesis problems which are affine with respect to unknown polynomials variables. The results of almost stability are also extended to simultaneous asymptotic stability at the expense of adding some extra affine polynomial inequalities. Numerical method for verification of positivity of multivariate polynomials based on sum of square decomposition is used. Numerical examples and a fault tolerant design scheme are used to show the effectiveness of the proposed methods.

IDA-PBC Algebraico Para Sistemas Polinomiales con Saturación en las Entradas: Un Enfoque Basado en SOS

The necessity to deal with partial differential equations (PDEs) and the dissipation condition are the mainadversities in the application of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). Recently,an algebraic solution of IDA-PBC has been explored for a class of affine polynomial systems by using sum of squares(SOS) and semidefinite programming (SDP). In this work, we extend the previous method by incorporating actuatorsaturation (AS) and two minimization objectives in the SDP. Our results are validated on two polynomial systems.

Azumaya loci and discriminant ideals of PI algebras

We prove that, under mild assumptions, for all positive integers ℓ, the zero set of the discriminant ideal Dℓ(R/Z(R),tr) of a prime affine polynomial identity (PI) algebra R coincides with the zero set of the modified discriminant ideal MDℓ(R/Z(R),tr) of R, and give an explicit description of this set in terms of the dimensions of the irreducible representations of R. Furthermore, we prove that, when ℓ is the square of the PI-degree of R, this zero set is precisely the complement of the Azumaya locus of R. This description is used to classify the Azumaya loci of the multiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime affine PI algebra R coincides with the singular locus of the center of R, provided that the discriminant ideal has height at least 2, R has finite global dimension and R is a Cohen–Macaulay module over its center.

IDA-PBC for Polynomial Systems: An SOS-based Approach

Algebraic solutions for the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) problem are introduced for a class of affine polynomial systems. These classes do not involve to solve partial differential equations for the matching condition. The proposed procedure leads to conditions that may be solved by using sum of squares (SOS) and semidefinite programming (SDP). Furthermore, some special parametrizations for the desired Hamiltonian function are analyzed and respective reductions into SOS inequalities are provided. Results are validated on a polynomial second order system and the well-known cart-pole system.

A vertex-based scheme on polyhedral meshes for advection–reaction equations with sub-mesh stabilization

We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection–reaction equations. Error estimates of order O(h3/2) are established in the discrete inf–sup stability norm which includes the mesh-dependent weighted advective derivative. The two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.

An algebraic geometry version of the Kakeya problem

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials f,g∈Fq0[x,y] and any Fq/Fq0, the image of the map Fq3→Fq3 given by (s,x,y)↦(s,sx+f(x,y),sy+g(x,y)) has size at least q34−O(q5/2) and prove the special case when f=f(x),g=g(y). We also prove it in the case f=f(y),g=g(x) under the additional assumption f′(0)g′(0)≠0 when f,g are both affine polynomials. Our approach is based on a combination of Cauchy–Schwarz and Lang–Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials.

Convex relaxations for L2-gain analysis of piecewise affine/polynomial systems

This paper proposes some sufficient conditions based on the computation of polynomial and piecewise polynomial storage functions for -gain analysis of discrete-time piecewise affine or piecewise polynomial systems. The computation of such storage functions is performed by means of convex optimisation techniques via the sum-of-squares decomposition of multivariate polynomials.

An Arithmetic Poisson Formula for the multi-variate resultant

In these pages we compute the expectation of several functions of multi-variate complex polynomials. The common thread of all our outcomes is the basic technique used in their proofs. The used techniques combine essentially the unitary invariance of Bombieri–Weyl’s Hermitian product and some elementary Integral Geometry. Using different combinations of these techniques we compute the expectation of the logarithm of the absolute value of an affine polynomial and we compute the expected value of Akatsuka Zeta Mahler’s measure. As main consequences of these results and techniques, we show a probabilistic answer to question (d) in Armentano and Shub (2012) [2], concerning the complexity of one point homotopy, and an Arithmetic Poisson Formula for the multi-variate resultant. These two last statements and bounds are related to the complexity of algorithms for polynomial equation solving.

Cell centered Galerkin methods for diffusive problems

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.

Predictive Control for Polynomial Systems Subject to State and Input Constraints

Abstract We present a nonlinear model predictive control (NMPC) approach to design optimal feedback controllers for the class of continuous-time input affine polynomial systems subject to state and input constraints. The corresponding feedback law is obtained via the solution of an efficient to solve convex optimization problem subject to sum of squares (SOS) constraints. To serve the conflicting requirements of performance and computational costs in various applications, the approach comes in three different control schemes which differ in how many times the (updated) optimization problem has to be solved, and whether this is done on- or offline. Each of the proposed control schemes minimizes an upper bound on the cost functional while guaranteeing closed-loop stability and satisfaction of input and state constraints. Finally, simulations of an example system show the applicability and the effectiveness of the proposed controllers.

Positional accuracy improvement: a comparative study in Shanghai, China

With the rapid development of geospatial data capture technologies such as the Global Positioning System, more and higher accuracy data are now readily available to upgrade existing spatial datasets having lower accuracy using positional accuracy improvement (PAI) methods. Such methods may not achieve survey-accurate spatial datasets but can contribute to significant improvements in positional accuracy in a cost-effective manner. This article addresses a comparative study on PAI methods with applications to improve the spatial accuracy of the digital cadastral for Shanghai. Four critical issues are investigated: (1) the choice of improvement model in PAI adjustment; five PAI models are presented, namely the translation, scale and translation, similarity, affine, and second-order polynomial models; (2) the choice of estimation method in PAI adjustment; three estimation methods in PAI adjustment are proposed, namely the classical least squares (LS) adjustment, which assumes that only the observation vector contains error, the general least squares (GLS) adjustment, which regards both the ground and map coordinates of control points as observations with errors, and the total least squares (TLS) adjustment, which takes the errors in both the observation vector and the design matrix into account; (3) the impact of the configuration of ground control points (GCPs) on the result of PAI adjustment; 12 scenarios of GCP configurations are tested, including different numbers and distributions of GCPs; and (4) the deformation of geometric shape by the above-mentioned transformation models is presented in terms of area and perimeter. The empirical experiment results for six test blocks in Shanghai demonstrated the following. (1) The translation model hardly improves the positional accuracy because it accounts only for the shift error within digital datasets. The other four models (i.e., the scale and translation, similarity, affine, and second-order polynomial models) significantly improve the positional accuracy, which is assessed at checkpoints (CKPs) by calculating the difference between the updated coordinates transformed from the map coordinates and the surveyed coordinates. On the basis of the refined Akaike information criterion, the two best optimal transformation models for PAI are determined as the scale and translation and affine transformation models. (2) The weighted sum of square errors obtained using the GLS and TLS methods are much less than those obtained using the classical least squares method. The result indicates that both the GLS and TLS estimation methods can achieve greater reliability and accuracy in PAI adjustment. (3) The configuration of GCPs has a considerable effect on the result of PAI adjustment. Thus, an optimal configuration scheme of GCPs is determined to obtain the highest positional accuracy in the study area. (4) Compared with the deformations of geometric shapes caused by the transformation models, the scale and translation model is found to be the best model for the study area.

Difference analysis and registration research of a pair of complementary cylindrical panoramic images

Based on the introduction of the complementary catadioptric omnidirectional imaging system,the difference between correspondence points in two complementary cylindrical panoramic images was analyzed in detail.For the coordinate difference of correspondence points in two images,Harris corner was chosen to search transform model of registration,to solve the registration problem of the two complementary cylindrical panoramic images.The experimental results cover the registration results,precision of using affine model,projective model and polynomial model for registration,which suggest that third-order polynomials model is more suitable.

Bounds on the degree of APN polynomials: the case of x −1 + g(x)

In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x ?1 + g(x) where g is any non $${\mathcal{F}_{2}}$$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ? 3 where these functions are equivalent to x 3.

$\boldsymbol{x^{2^l+1}+x+a}$ and related affine polynomials over GF (2 k )

In this paper, the polynomials \(P_a(x)=x^{2^l+1}+x+a\) with a ∈ GF(2k) are studied. New criteria for the number of zeros of Pa(x) in GF(2k) are proved. We also study the affine polynomial \(a^{2^l}x^{2^{2l}}+x^{2^l}+ax+1\) which is closely related to Pa(x). In many cases, explicit expressions for calculating zeros of these polynomials are provided.