A chain rule for the resultant of two polynomials

Archiv der Mathematik -     

A resultant condition for the irreducibility of the polynomials

We prove a criterion for the irreducibility of the polynomials in one indeterminate with the coefficients in the valuation ring of a discrete valued field. From this result we deduce the Schönemann, Eisenstein, and Akira irreducibility criteria. The results obtained can also be used for proving that some polynomials in several indeterminates are irreducible.     

Algorithms for finding the minimal polynomials and inverses of resultant matrices

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Gröbner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.     

Several polynomials associated with the harmonic numbers

We develop polynomials in z∈C for which some generalized harmonic numbers are special cases at z=0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials.     

Several classes of complete permutation polynomials

In this paper, three classes of monomials and one class of trinomials over finite fields of even characteristic are proposed. They are proved to be complete permutation polynomials.     

Multivariate versions and two problems of Schoenberg

Translated from:Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, 1989, pp. 72–79.     

Relationship between the zeros of two polynomials

In this paper, we shall follow a companion matrix approach to study the relationship between zeros of a wide range of pairs of complex polynomials, for example, a polynomial and its polar derivative or Sz.-Nagy’s generalized derivative. We shall introduce some new companion matrices and obtain a generalization of the Weinstein-Aronszajn Formula which will then be used to prove some inequalities similar to Sendov conjecture and Schoenberg conjecture and to study the distribution of equilibrium points of logarithmic potentials for finitely many discrete charges. Our method can also be used to produce, in an easy and systematic way, a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial.     

On absolute continuity of the spectrum of random polynomials orthogonal on the circle and their continual versions

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 194, pp. 170–173, 1992.     

Common multiples and common divisors of matrix polynomials, II. Vandermonde and resultant matrices

In this paper explicit formulas are given for least common multiples and greatest common divisors of a finite number of matrix polynomials in terms of the coefficients of the given polynomials. An important role is played by block matrix generalizations of the classical Vandermonde and resultant matrices. Special attention is given to the evaluation of the degrees and other characteristics. Applications to matrix polynomial equations and factorization problems are made.     

Orthogonal polynomials in two variables

Summary Some properties of orthogonal (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those linear homogeneous partial differential eqnations of the form L[w]+λw=0, having a set of polynomials as solution, are characterized; and a detailed study is made of all such equations of second order whose polynomial solutions form an orthogonal (or generalized orthogonal) set. Supported byN.S.F. Grant GP-5311.     

Several versions of the devour digest tidy-up heuristic for unconstrained binary quadratic problems

The unconstrained binary quadratic minimization problem is known to be NP-hard and due to its computational challenge and application capability, it becomes more and more considered and involved by the recent research studies, including both exact and heuristic solution approaches. Our work is in line with what was suggested by Glover et al. (in Eur. J. Oper. Res. 137, 272–287, 2002) who proposed one pass heuristics as alternatives to the well-known Devour Digest Tidy-up procedure (DDT) of Boros et al. (in RRR 39-89, 1989). The “devour” step sets a term of the current representation to 0 or 1, and the “tidy-up” step substitutes the logical consequences derived from the “digest” step into the current quadratic function. We propose several versions of the DDT constructive heuristic based on the alternative representation of the quadratic function. We also present an efficient implementation of local search using one-flip and two-flip moves that simultaneously change the values of one or two binary variables. Computational experiments performed on instances up to 7000 variables show the efficiency of our implementation in terms of quality improvement and CPU use enhancement.     

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.     

Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables

We introduce polynomials $B^n_{k}(\boldmath{x};\omega|q)$ of total degree n, where $\boldmath{k} = (k_1,\ldots,k_d)\in\mathbb N_0^d, \; 0\le k_1+\ldots+k_d\le n$ , and $\boldmath{x}=(x_1,x_2,\ldots,x_d)\in\mathbb R^d$ , depending on two parameters q and ω, which generalize the multivariate classical and discrete Bernstein polynomials. For ω=0, we obtain an extension of univariate q-Bernstein polynomials, introduced by Phillips (Ann Numer Math 4:511–518, 1997). Basic properties of the new polynomials are given, including recurrence relations, q-differentiation rules and de Casteljau algorithm. For the case d=2, connections between $B^n_{k}(\boldmath{x};\omega|q)$ and bivariate orthogonal big q-Jacobi polynomials—introduced recently by the first two authors—are given, with the connection coefficients being expressed in terms of bivariate q-Hahn polynomials. As limiting forms of these relations, we give connections between bivariate q-Bernstein and Dunkl’s (little) q-Jacobi polynomials (SIAM J Algebr Discrete Methods 1:137–151, 1980), as well as between bivariate discrete Bernstein and Hahn polynomials.     

On the norm of polynomials of two adjoint projections

Letf(X, Y) be a polynomial of two non-commuting variables and letP be an arbitrary nontrivial projection operator in Hilbert space. The class of all polynomialsf(X, Y) for which f(P, P ^*) depends only onf and P are described. In the case when such a dependence exists the explicit formula is obtained. Some applications to singular integral operators are given.     

Semiclassical orthogonal polynomials in two variables

In this work, semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix Pearson-type differential equation. Semiclassical functionals are characterized by means of the analogue of the structure relation in one variable. Moreover, non trivial examples of semiclassical orthogonal polynomials in two variables are given.