The space of polynomials with roots of bounded multiplicity

数理解析研究所講究録, 1993

2007

The aim of this paper is to study many interpolation problems in the space of polynomials of w-degree n. In order to do this, some new results concerning the polynomial spaces of w-degree n are given. We consider only the case of functions in two variables, but all the results obtained can be easily extended to many variables. We found a set of conditions for which, Π n,w , the space of polynomials of w-degree is an interpolation space. More details are obtained for the weight w = (1, w 2 ).

arXiv (Cornell University), 2022

For each pair (m, n) of positive integers with (m, n) = (1, 1) and an arbitrary field F with algebraic closure F, let Poly d,m n (F) denote the space of m-tuples (f 1 (z), • • • , f m (z)) ∈ F[z] m of F-coefficients monic polynomials of the same degree d such that the polynomials {f k (z)} m k=1 have no common root in F of multiplicity ≥ n. These spaces Poly d,m n (F) were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts. In previous we determined explicitly the homotopy type of this space in the case F = C. In this paper, we investigate the case F = R.

MATHEMATICA SCANDINAVICA, 2009

For any polynomial $P\in {\mathsf C} [X_1,X_2,\ldots,X_n]$, we describe a $\mathsf C$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$ is the number of irreducible factors of $P$. Moreover, the knowledge of $F(P)$ gives a complete factorization of the polynomial $P$ by taking gcd's. This generalizes previous results by Ruppert and Gao in the case $n=2$.

Results in Mathematics, 2015

Assume that a linear space of real polynomials in d variables is given which is translation and dilation invariant. We show that if a sequence in this space converges pointwise to a polynomial, then the limit polynomial belongs to the space, too.

Journal of Approximation Theory, 1987

Journal of the Korean Mathematical Society, 2009

We give conditions so that a polynomial be factorable through an L r (µ) space. Among them, we prove that, given a Banach space X and an index m, every absolutely summing operator on X is 1-factorable if and only if every 1-dominated m-homogeneous polynomial on X is right 1-factorable, if and only if every 1-dominated m-homogeneous polynomial on X is left 1-factorable. As a consequence, if X has local unconditional structure, then every 1-dominated homogeneous polynomial on X is right and left 1-factorable. We give conditions so that a homogeneous polynomial P between Banach spaces be factorable through an L r (µ)-space, either in the form P = Q • T , where T is a (linear) operator and Q is a polynomial (right r-factorization), or in the form P = T • Q (left r-factorization). It is shown in particular that, given a Banach space X and an index m, every absolutely summing operator on X is 1-factorable if and only if every 1dominated m-homogeneous polynomial on X is right 1-factorable, if and only if every 1-dominated m-homogeneous polynomial on X is left 1-factorable. As a consequence, if X has local unconditional structure, then every 1-dominated m-homogeneous polynomial on X is right and left 1-factorable. Throughout, X, Y , Z denote Banach spaces, X * is the dual of X, and B X stands for its closed unit ball. The closed unit ball B X * will always be endowed with the weak-star topology. By N we represent the set of all natural numbers, and by K the scalar field (real or complex). We use the symbol L(X, Y) for the space of all (linear bounded) operators from X into Y endowed with the operator norm. Given a space Y we shall denote by k Y the natural embedding of Y into its bidual Y * *. Given m ∈ N, we denote by P(m X, Y) the space of all m-homogeneous (continuous) polynomials from X into Y endowed with the supremum norm. Recall that with each P ∈ P(m X, Y) we can associate a unique symmetric

Journal of Mathematical Analysis and Applications, 2005

We prove that a majorization-type relation among the root sets of three polynomials implies that the same relation holds for the root sets of their derivatives. We then use this result to give a unified derivation of the classical results due to Sz.-Nagy, Robinson, Meir and Sharma which relate the span of a polynomial to the spans of its first or higher derivatives. We also show how this relation can be generated by interlacing polynomials.  2005 Elsevier Inc. All rights reserved.

Ukrainian Mathematical Journal

The criterion for the denseness of polynomials in the space L_2 ( R , d μ ) established by Hamburger in 1921 is extended to the spaces L_p ( R , d μ ) , 1 ≤ p < ∞ .

2001

For a Banach space E we define the class PK(E) of K-bounded N -homogeneous polynomials, where K is a bounded subset of E′. We investigate properties of K which relate the space PK(E) with usual subspaces of P(E). We prove that K-bounded polynomials are approximable when K is a compact set where the identity can be uniformly approximated by finite rank operators. The same is true when K is contained in the absolutely convex hull of a weakly null basic sequence of E′. Moreover, in this case we prove that every K-bounded polynomial is extendible to any larger space.

Let $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$ be a $C^k$ curve of monic polynomials, $a_i \in C^k(I,\mathbb C)$ where $I \subset \mathbb R$ is an interval. We show that if $k=k(n)$ is sufficiently large then any choice of continuous roots of $P_a$ is locally absolutely continuous, in a uniform way with respect to $\max_j \|a_j\|_{C^k}$ on compact subintervals. This solves a problem that was open for more then a decade and shows that some systems of pseudodifferential equations are locally solvable. Our main tool is the resolution of singularities.

Mediterranean Journal of Mathematics, 2014

In this paper some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization we introduce a numerical quantity depending on the variety of the local polynomial only. Moreover, we show that the known characterization of polynomials among generalized polynomials can be simplified: a generalized polynomial is a polynomial if and only if its variety contains finitely many linearly independent additive functions.

Ukrainian Mathematical Journal, 2005

Pacific Journal of Mathematics, 1986

Mathematische Nachrichten, 2016

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.