Polynomials Of Arbitrary Degree Research Articles

Imaginary projections of polynomials

We introduce the imaginary projection of a multivariate polynomial f∈C[z] as the projection of the variety of f onto its imaginary part, I(f)={Im(z):z∈V(f)}. Since a polynomial f is stable if and only if I(f)∩R>0n=∅, the notion offers a novel geometric view underlying stability questions of polynomials.We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

Nodal finite element de Rham complexes

We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom (DoFs) and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical N\'{e}d\'{e}lec and Brezzi-Douglas-Marini (BDM) finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for ${H}(\mathrm{div})$ with regularity $r=2$ coincide with the nonstandard elements given by Stenberg (Numer Math 115(1): 131-139, 2010). We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity. The construction of the new families is motivated by the finite element systems.%, and we also establish local exact sequences (geometric decomposition) for the new elements.

Orthogonal-polynomials-based integral inequality and its applications to systems with additive time-varying delays

Recently, a polynomials-based integral inequality was proposed by extending the Moon’s inequality into a generic formulation. By imposing certain structures on the slack matrices of this integral inequality, this paper proposes an orthogonal-polynomials-based integral inequality which has lower computational burden than the polynomials-based integral inequality while maintaining the same conservatism. Further, this paper provides notes on relations among recent general integral inequalities constructed with arbitrary degree polynomials. In these notes, it is shown that the proposed integral inequality is superior to the Bessel–Legendre (B–L) inequality and the polynomials-based integral inequality in terms of the conservatism and computational burden, respectively. Moreover, the effectiveness of the proposed method is demonstrated by an illustrative example of stability analysis for systems with additive time-varying delays.

Fast operations on linearized polynomials and their applications in coding theory

This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial degree s, independent of the underlying field extension degree m. We show that our multiplication algorithm is faster than all known ones when s≤m. Using a result by Caruso and Le Borgne (2017), this immediately implies a sub-quadratic division algorithm for linearized polynomials for arbitrary polynomial degree s. Also, we propose algorithms with sub-quadratic complexity for the q-transform, multi-point evaluation, computing minimal subspace polynomials, and interpolation, whose implementations were at least quadratic before. Using the new fast algorithm for the q-transform, we show how matrix multiplication over a finite field can be implemented by multiplying linearized polynomials of degrees at most s=m if an elliptic normal basis of extension degree m exists, providing a lower bound on the cost of the latter problem. Finally, it is shown how the new fast operations on linearized polynomials lead to the first error and erasure decoding algorithm for Gabidulin codes with sub-quadratic complexity.

Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method

In this paper, we present for the first time guaranteed upper bounds for the staggered discontinuous Galerkin method for diffusion problems. Two error estimators are proposed for arbitrary polynomial degrees and provide an upper bound on the energy error of the scalar unknown and $$L^2$$ -error of the flux, respectively. Both error estimators are based on the potential and flux reconstructions. The potential reconstruction is given by a simple averaging operator. The equilibrated flux reconstruction can be found by solving local Neumann problems on elements sharing an edge with a Raviart–Thomas mixed method. Reliability and efficiency of the two a posteriori error estimators are proved. Numerical results are presented to validate the theoretical results.

Non-minimal order low-frequency H∞ filtering for uncertain discrete-time systems

This paper proposes a sufficient condition for the discrete-time robust H∞ filtering design problem with low-frequency specifications using an extension of the generalized Kalman-Yakubovich-Popov lemma. The matrices of the system are supposed to be uncertain, time-invariant and to belong to a polytopic domain. The proposed approach takes advantage of a non-minimal filter structure, that is, a filter with order greater than the order of the system being filtered, to provide improved H∞ bounds for low-frequency specifications. The condition can be solved by means of linear matrix inequality relaxations with slack variables and Lyapunov matrices which are considered as homogeneous polynomials of arbitrary degree. Numerical examples illustrate the improvements on the H∞ bounds provided by the non-minimal filter structure in combination with the more accurate polynomial approximations (higher degrees) for the optimization variables.

Orthogonal projectors onto spaces of periodic splines

The main result of this paper is a proof that for any integrable function f on the torus, any sequence of its orthogonal projections (P˜nf) onto periodic spline spaces with arbitrary knots Δ˜n and arbitrary polynomial degree converges to f almost everywhere with respect to the Lebesgue measure, provided the mesh diameter ∣Δ˜n∣ tends to zero. We also give a new and simpler proof of the fact that the operators P˜n are bounded on L∞ independently of the knots Δ˜n.

Unsymmetric multi-level hanging nodes and anisotropic polynomial degrees in [formula omitted]-conforming higher-order finite element methods

The implementation of higher-order finite element schemes that can handle multi-level hanging nodes is known to be a difficult task. In fact, most of the available literature on hanging nodes in finite element schemes restricts to one-level hanging nodes resulting from symmetric bisections. The intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a H1-conforming higher-order finite element method. The corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric, multi-level hanging nodes and arbitrary anisotropic polynomial degree distributions, where special care is given to possible orientation problems. The meshes may even result from non-matching refinements. Given these data structures and algorithms, an extension for Serendipity spaces is described in detail along with some other techniques to improve computational efficiency. Numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher-order finite element schemes have to offer. To the best of our knowledge, this paper offers the most comprehensive numerical results so far for various three-dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as (multi-level) hanging nodes. Most notably, the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates. However, the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen (full) tensor product or Serendipity finite element spaces. In addition to these numerical results, we find that systems of equations defined via finite element spaces using the nodal Lagrange polynomials with Gauss–Lobatto quadrature points as support points yield drastically improved condition numbers.

Newton's method in practice: Finding all roots of polynomials of degree one million efficiently

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm that can be applied routinely to find all roots without deflation and with the inherent numerical stability of Newton's method.We specify an algorithm that provably terminates and finds all roots of any polynomial of arbitrary degree, provided all roots are distinct and exact computation is available. It is known that Newton's method is inherently stable, so computing errors do not accumulate; we provide an exact bound on how much numerical precision is sufficient.

An improvement to the Hasse–Weil bound and applications to character sums, cryptography and coding

The Hasse–Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse–Weil bound for the curve defined by yp−y=f(x) over the finite field Fq, where p is the characteristic of Fq. In 1993, Moreno–Moreno [7] gave an improvement to the Hasse–Weil bound for this family of curves. Recently, Kaufman and Lovett [4, FOCS2011] showed that the Hasse–Weil bound can be improved for this family of curves with f(x)=g(x)+h(x), where g(x) is a polynomial of degree ≪q and h(x) is a sparse polynomial of arbitrary degree but bounded weight degree. The other recent improvement by Rojas-Leon and Wan [9, Math. Ann. 2011] shows that an extra p can be removed for this family of curves if p is very large compared with polynomial degree of f(x) and logp⁡q.In this paper, we focus on the most interesting case for applications, namely p=2. We show that the Hasse–Weil bound for this special family of curves can be improved if q=2n with odd n⩾3 which is the same case where Serre [10] improved the Hasse–Weil bound. However, our improvement is greater than Serre's and Moreno–Morenao's improvements for this special family of curves. Furthermore, our improvement works for p=2 compared with the requirement of large p by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse–Witt invariant 0 and we show that the Hasse–Weil bound can be improved for any curves with the Hasse–Witt invariant 0. The main tool used in our proof involves Newton polygon and some results in algebraic geometry.

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

Here I present a method how intersections of a certain density matrix of rank two with the zero-polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely in terms of the Boch-sphere analogy, solves this problem immediately. To this end, I determine the zero-polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero-polytope (zero-simplex) are exact values. I apply this procedure to a superposition of the four qubit GHZand W-state. It can however be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.

High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

We present a new AFEM for the Laplace---Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $$W^1_\infty $$Wź1 and piecewise in a suitable Besov class embedded in $$C^{1,\alpha }$$C1,ź with $$\alpha \in (0,1]$$źź(0,1]. The idea is to have the surface sufficiently well resolved in $$W^1_\infty $$Wź1 relative to the current resolution of the PDE in $$H^1$$H1. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in $$W^1_\infty $$Wź1 and PDE error in $$H^1$$H1.

A New Generalization of the P 1 Non-Conforming FEM to Higher Polynomial Degrees

Abstract This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

A convenient expression of the time-derivative , of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z, and solvable dynamical systems

Let pN (z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let zn ≡ zn (t) be its N zeros: . In this paper we report a convenient expression of the k-th time-derivative of the zero zn (t). This formula plays a key role in the identification of classes of solvable dynamical systems describing the motion of point-particles moving in the complex z-plane while nonlinearly interacting among themselves; one such example, featuring many arbitrary parameters, is reported, including its variation describing the motion of many particles moving in the real Cartesian xy-plane and interacting among themselves via rotation-invariant Newtonian equations of motion (”accelerations equal forces”).

On some properties of the Abel–Goncharov polynomials and the Casas-Alvero problem

ABSTRACTWe derive new properties of the Abel–Goncharov interpolation polynomials, relating them to investigate necessary and sufficient conditions for an arbitrary polynomial of degree n to be trivial, i.e. to have the form . These results are associated with an open problem, conjectured in 2001 by E. Casas- Alvero. It says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. In particular, we establish determinantal representation of the Abel–Goncharov interpolation polynomials, having its own interest. Among other results are new Sz.-Nagy-type identities for complex roots and a generalization of the Schoenberg conjectured analog of Rolle's theorem for polynomials with real and complex coefficients.

Information matrix and D-optimal design with Gaussian inputs for Wiener model identification

We present a closed form expression for the Fischer’s information matrix associated with the identification of Wiener models. In the derivation we assume that the input signal is Gaussian. The analysis allows the linear sub-system in the Wiener model to have a generic rational transfer function of arbitrary order. It also allows the static nonlinearity of the Wiener model to be a polynomial of arbitrary degree. In addition, we show how this analysis can be used to design tractable algorithms for D-optimal input design. The idea is further extended to design optimal inputs consisting of a sequence of Gaussian signals with different mean values and variances. By combining Gaussian inputs with different means we can tune the amplitude distribution of the input to achieve the best identification accuracy in D-optimal sense. The analytical results are also illustrated with some numerical simulations.

Bivariate fully fuzzy interpolation problem using artificial neural networks approach

Artificial neural networks modeling is one of the most prominent techniques for solving more complicated math- ematical problems that can not be solved in the traditional computing environments. The work described here intends to offer an efficient bivariate fuzzy interpolation methodology based on the artificial neural networks approach. It has several notable features including high processing speeds and the ability to learn the solution to a problem from a set of examples which categorizes them in line of intelligent systems. To do this, a multilayer feed-forward neural architecture is depicted for constructing a fully fuzzy interpolating polynomial of arbitrary degree. Then, a back-propagation supervised learning optimization algorithm will be applied for estimating the unknown fuzzy coefficients of the solution polynomial. Finally, the advantage of our technique is illustrated by using some practical examples to show the ability of the improved algorithm in solving rigorous problems.

Irreducibility criteria for compositions of polynomials with integer coefficients

We provide irreducibility criteria for some classes of compositions of polynomials with integer coefficients of the form $$F\circ G$$ , with F being a quadratic irreducible polynomial and G a polynomial of arbitrary degree.

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L2 norm. We then derive optimal a priori error estimates in the H1 and L2 norm for a FEM with variational crimes due to numerical integration. As an application, we derive a priori error estimates for a numerical homogenization method applied to nonlinear monotone elliptic problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 955–969, 2016