An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions

We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of s polynomials in R[X ₁, …,X _k ] whose degrees are at most d is bounded by

On computation of the greatest common divisor of several polynomials over a finite field

We propose a probabilistic algorithm to reduce computing the greatest common divisor of m polynomials over a finite field (which requires computing m−1 pairwise greatest common divisors) to computing the greatest common divisor of two polynomials over the same field.     

Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.     

Pseudo-Orthogonal Polynomials

We consider the classical problem of transforming an orthogonality weight of polynomials by means of the space R _n . We describe systems of polynomials called pseudo-orthogonal on a finite set of n points. Like orthogonal polynomials, the polynomials of these systems are connected by three-term relations with tridiagonal matrix which is nondecomposable but does not enjoy the Jacobi property. Nevertheless these polynomials possess real roots of multiplicity one; moreover, almost all roots of two neighboring polynomials separate one another. The pseudo-orthogonality weights are partly negative. Another result is the analysis of relations between matrices of two different orthogonal systems which enables us to give explicit conditions for existence of pseudo-orthogonal polynomials.     

Solutions of Schrödinger Equations on Compact Lie Groups via Probabilistic Methods

By means of probabilistic methods global solutions of Schrödinger equations on compact connected semisimple Lie groups are constructed. The potentials and initial conditions are taken from the algebra of trigonometric polynomials. For these the solutions exist in the L ²-sense and pointwise.     

An expansion theorem concerning Wilson functions and polynomials

We prove that a relatively general even function f(x) (satisfying a vanishing condition, and also some analyticity and growth conditions) on the real line can be expanded in terms of a certain function series closely related to the Wilson functions introduced by Groenevelt in 2003. The coefficients in the expansion of f will be inner products in a suitable Hilbert space of f and some polynomials closely related to Wilson polynomials (these are well-known hypergeometric orthogonal polynomials).     

The bilateral birth–death chain generated by the associated Jacobi polynomials

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one-step transition probability matrix of a discrete-time bilateral birth–death chain with state space on $\mathbb{Z}$. We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.     

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.     

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.     

Criteria for the constant sign property for real polynomials on a segment

It is well known that classic theorems of Markov and Lukach for real polynomials which have a constant sign on a segment are ineffective. In this paper we obtain criteria for the constant sign property on a segment for real polynomials of the fourth degree and formulate certain their generalizations. The mentioned criteria are stated in terms of the coefficients of the polynomials under consideration.     

On real and complex-valued bivariate Chebyshev polynomials

In the real uniform approximation of the function x^myⁿ by the space of bivariate polynomials of total degree m + n − 1 on the unit square, the product of monic univariate Chebyshev polynomials yields an optimal error. We exploit the fundamental Noether's theorem of algebraic curves theory to give necessary and sufficient conditions for unicity and to describe the set of optimal errors in case of nonuniqueness. Then, we extend these results to the complex approximation on biellipses. It turns out that the product of Chebyshev polynomials also provides an optimal error and that the same kind of uniqueness conditions prevail in the complex case. Yet, when nonuniqueness occurs, the characterization of the set of optimal errors presents peculiarities, compared to the real problem.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Generic extensions and generic polynomials for semidirect products

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group AG over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension.Generic polynomials for several small groups are constructed.     

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product with and (x)is a weight function.We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to (x) and we find the recurrence relation verified by such a family. If the weight is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.     

Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99–107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to S _p or A _p .     

Bivariate Factorizations via Galois Theory, with Application to Exceptional Polynomials

We present a method for factoring polynomials of the shapef(X) − f(Y), wherefis a univariate polynomial over a fieldk. We then apply this method in the case whenfis a member of the infinite family of exceptional polynomials we discovered jointly with H. Lenstra in 1995; factoringf(X) − f(Y) in this case was posed as a problem by S. Cohen shortly after the discovery of these polynomials.     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

Chebyshev polynomials are not always optimal

We are concerned with the problem of finding the polynomial with minimal uniform norm on among all polynomials of degree at most n and normalized to be 1 at c. Here, is a given ellipse with both foci on the real axis and c is a given real point not contained in . Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this work, we show that this is not true in general. Moreover, we derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.