The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree

One of the considerable discussions in data interpolation is to find the optimal number of data which minimizes the error of the interpolation polynomial. In this paper, first the theorems corresponding to the equidistant nodes and the roots of the Chebyshev polynomials are proved in order to estimate the accuracy of the interpolation polynomial, when the number of data increases. Based on these theorems, then we show that by using a perturbation method based on the CESTAC method, it is possible to find the optimal degree of the interpolation polynomial. The results of numerical experiments are presented.     

Estimates for the orders of zeros of polynomials in some analytic functions

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.     

Approximation by Bernstein–Durrmeyer-type operators in compact disks

In this paper, the order of simultaneous approximation and Voronovskaja-type theorems with quantitative estimate for complex Bernstein–Durrmeyer-type polynomials attached to analytic functions on compact disks are obtained. Our results show that extension of the complex Bernstein–Durrmeyer-type polynomials from real intervals to compact disks in the complex plane extends approximation properties.     

Matrix polynomials,roots and spectral factors

This paper discusses some properties of matrix polynomials and a computational procedure for finding the matrix roots of such polynomials and their relationship to spectral factorization. Polynomials of order n with square matrix coefficients of order N are considered. The computational procedure is of interest in the analysis and design of multivariable control systems.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

More zeros of krawtchouk polynomials

Three theorems are given for the integral zeros of Krawtchouk polynomials. First, five new infinite families of integral zeros for the binary (q = 2) Krawtchouk polynomials are found. Next, a lower bound is given for the next integral zero for the degree four polynomial. Finally, three new infinite families inq are found for the degree three polynomials. The techniques used are from elementary number theory.     

p-Adic valuations associated to fibers and images of polynomial functions

The Chevalley-Warning and the Ax-Katz theorems, which provide the p-adic valuations for the number of solutions of the system of polynomials in finite fields, are stated in terms of the degrees of the associated polynomials. Recently, Clark et al. reformulated the theorems in terms of the cardinalities of fibers and images of the associated polynomial functions, and raised a question in this setting. In this paper, we give a negative answer to that question and improve their results in general case using p-adic lifting via the Teichmüller character.     

On characterizations of classical polynomials

It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients, Rodrigues formula, etc. In this paper we present a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the “q-Hahn tableau”. Finally, a detailed discussion of a characterization by Marcellán et al. is presented.     

Roots and polynomials as Homeomorphic spaces

We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree n and the space of n roots with suitable topologies, we are able to formulate the classical theorems in the form of a homeomorphism. Related topological facts are also considered.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F-polynomials j_∑fj−1(j(x−1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1,1), and present ways to construct simplicial complexes with Schur stable F-polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.     

一元多项式学习中易犯的几个错误

结合一元多项式中的一些重要概念,如多项式的最大公因式、多项式的重根及不可约多项式等,分析一元多项式学习中易犯的错误,并强调运用定理时要注意其适用的条件和前提．     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.     

Approximation by complex q-Durrmeyer polynomials in compact disks

In this paper, the order of approximation and Voronovskaja type results with quantitative estimate for complex q-Durrmeyer polynomials attached to analytic functions on compact disks are obtained.     

Combinatorics & Topology of Stratifications of The Space of Monic Polynomials With Real Coefficients

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity k. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.     

Asymptotic constants for the error of Hermite-Fejér interpolation on the unit circle

The paper deals with the rate of convergence for the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system the n roots of a complex number with modulus one. The order of convergence and the asymptotic constants are obtained when we consider analytic functions on open disks and open annulus containing the unit circle.     

On properties of difference polynomials

We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f (z)f (z + c).     

Harmonic polynomials, hyperspherical harmonics, and atomic spectra

The properties of monomials, homogeneous polynomials and harmonic polynomials in d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection. Many important properties of spherical harmonics, Gegenbauer polynomials and hyperspherical harmonics follow from these formulas. Harmonic projection also provides alternative ways of treating angular momentum and generalised angular momentum. Several powerful theorems for angular integration and hyperangular integration can be derived in this way. These purely mathematical considerations have important physical applications because hyperspherical harmonics are related to Coulomb Sturmians through the Fock projection, and because both Sturmians and generalised Sturmians have shown themselves to be extremely useful in the quantum theory of atoms and molecules.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.