An analogue of the big <Emphasis Type="Italic">q</Emphasis>-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.     

Bivariate orthogonal polynomials in the Lyskova class

Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equation of the same type.     

To solving multiparameter problems of algebra. 9. The Ψ<Emphasis Type="Italic">F-q</Emphasis> method for factorizing invariant polynomials and its applications

A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173.     

Bounds for resultants of univariate and bivariate polynomials

The paper considers bounds on the size of the resultant for univariate and bivariate polynomials. For univariate polynomials we also extend the traditional representation of the resultant by the zeros of the argument polynomials to formal resultants, defined as the determinants of the Sylvester matrix for a pair of polynomials whose actual degree may be lower than their formal degree due to vanishing leading coefficients. For bivariate polynomials, the resultant is a univariate polynomial resulting by the elimination of one of the variables, and our main result is a bound on the largest coefficient of this univariate polynomial. We bring a simple example that shows that our bound is attainable and that a previous sharper bound is not correct.     

Die Polynomlösungen spezieller partieller Differentialgleichungen

One of the main problems in the theory of orthogonal polynomials in several variables is the determination of partial differential equations which have the given polynomials as their solutions. In this note, we consider partial differential equations which are two-dimensional generalizations of the classical differential equation for the Chebyshev polynomials in one variable and we will give conditions for its polynomial solutions. In addition, we will be able to determine all polynomials of a given class which are solutions of the partial differential equation under consideration. In the last section, we establish a connection between the different polynomial solutions.     

分块平方和分解

本文定义了分块平方和可分解多项式的概念.粗略地说,它是这样一类多项式,只考虑其支撑集(不考虑系数)就可以把它的平方和分解问题等价地转换为较小规模的同类问题(换句话说,相应的半正定规划问题的矩阵可以分块对角化).本文证明了近年文献中提出的两类方法—分离多项式(split polynomial)和最小坐标投影(minimal coordinate projection)—都可以用分块平方和可分解多项式来描述,证明了分块平方和可分解多项式集在平方和多项式集中为零测集.     

Khachiyan's linear programming algorithm

L. G. Khachiyan's polynomial time algorithm for determining whether a system of linear inequalities is satisfiable is presented together with a proof of its validity. The algorithm can be used to solve linear programs in polynomial time.     

Computation of the GCD of polynomials of several variables on a MPC

The article is devoted to computer calculations with polynomials of several variables, in particular, to the construction of sets of large-block polynomial operations and to the concretization of the generalized algorithm of Euclid for computing the greatest common divisor (GCD) of two polynomials of several variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 148–155, 1976.     

Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0,2π] and a Bernstein–Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein–Szegö polynomials. When the Bernstein–Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.     

Amortized multi-point evaluation of multivariate polynomials

The evaluation of a polynomial at several points is called the problem of multi-point evaluation. Sometimes, the set of evaluation points is fixed and several polynomials need to be evaluated at this set of points. Several efficient algorithms for this kind of “amortized” multi-point evaluation have been developed recently for the special cases of bivariate polynomials or when the set of evaluation points is generic. In this paper, we extend these results to the evaluation of polynomials in an arbitrary number of variables at an arbitrary set of points. We prove a softly linear complexity bound when the number of variables is fixed. Our method relies on a novel quasi-reduction algorithm for multivariate polynomials, that operates simultaneously with respect to several orderings on the monomials.     

Multivariate Markov polynomial inequalities and Chebyshev nodes

This article considers the extension of V.A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

To solving multiparameter problems of algebra. 8. The <Emphasis Type="Italic">RP-q</Emphasis> method and its applications

A new method (the RP-q method) for factorizing scalar polynomials in q variables and q-parameter polynomial matrices (q ≥ 1) of full rank is suggested. Applications of the algorithm to solving systems of nonlinear algebraic equations and some spectral problems for a q-parameter polynomial matrix F (such as separation of the eigenspectrum and mixed spectrum of F, computation of bases with prescribed spectral properties of the null-space of polynomial solutions of F, and computation of the hereditary polynomials of F) are considered. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 149–164.     

Stochastic localization of roots of random polynomials

A procedure is proposed for stochastic localization of the roots of polynomials whose coefficients are random variables with a joint distribution density. The results are applied to problems of solvability of polynomial formulas.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 131–134, 1988.     

Some characteristic properties of the weighted particular Schur polynomial mean

This paper provides some characteristic properties of the weighted particular Schur polynomial mean of several variables. In addition, an elementary proof of an important inequality involving the weighted particular Schur polynomial mean is given. Various related results involving a family of the Schur polynomials, symmetric polynomials, and other associated polynomials, together with the potential for their applications, are also considered.     

Jucys–Murphy elements,orthogonal matrix integrals,and Jack measures

We study symmetric polynomials whose variables are odd-numbered Jucys–Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.     

Green's function,Jensen measures,and bounded point evaluations

A compactly supported measure μ on the complex plane C is called a Jensen measure for 0 if $\text{log}\text{¦P(0)¦ ? ∝}\text{log}\text{¦P(z)¦dμ(z)}$ for every polynomial P. H²(μ) denotes the closure of the polynomials in L²(μ). We obtain the result that if μ is not the point mass at 0, then the functions in H²(μ) are analytic on an open set which contains 0 and whose closure contains the support of μ. The primary tool used to obtain this result is a generalized Green's function for a measure, and we also derive some of its properties.     

Orthogonal polynomials of several discrete variables associated with negative polynomial distribution

A system of orthogonal polynomials of several discrete variables associated with the negative polynomial distribution is constructed and analyzed. The explicit form of the polynomials and the generating function are obtained.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 46–51, 1988.     

To solving multiparameter problems of algebra. 10. Computing zeros of a scalar polynomial

The paper considers the problem of computing zeros of scalar polynomials in several variables. The zeros of a polynomial are subdivided into the regular (eigen-and mixed) zeros and the singular ones. An algorithm for computing regular zeros, based on a decomposition of a given polynomial into a product of primitive polynomials, is suggested. The algorithm is applied to solving systems of nonlinear algebraic equations. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 119–130.