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.Comment: 16 page

Perturbations of roots under linear transformations of polynomials

Let \cP_n be the complex vector space of all polynomials of degree at most $n$. We give several characterizations of the linear operators T\in\cL(\cP_n) for which there exists a constant $C > 0$ such that for all nonconstant p\in\cP_n there exist a root $u$ of $p$ and a root $v$ of $Tp$ with $|u-v|\leq C$. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of $p$ and $Tp$, the roots are never displaced by more than a uniform constant independent on $p$. We show that such ``good'' operators $T$ are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of $T$ for the relevant constants.Comment: 23 page

Best Approximations in Preduals of Von Neumann Algebras

This paper characterises the semi-Chebychev subspaces of preduals of von Neumann algebras. As an application it generalises the theorem of Doob, that says that H01 has unique best approximations in L1(T), to a large class of preannihilators of natural non-selfadjoint operator algebras including the nest algebras. Then it studies the semi-Chebychev subspaces of the trace class operators and shows that the only Chebychev *-diagrams are ‘triangular

Stability of roots of polynomials under linear combinations of derivatives

Let $T=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n$, where $D$ is the differentiation operator and $\alpha_0\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-\alpha_1/\alpha_0$.Comment: 18 pages, 4 figure

On the Location of Critical Points of Polynomials

Given a polynomial p of degree n ≥ 2 and with at least two distinct roots let Z(p) = { z: p(z) = 0}. For a fixed root α ∈ Z(p) we define the quantities ω(p, α) := min (formula) and (formula). We also define ω (p) and τ (p) to be the corresponding minima of ω (p,α) and τ (p,α) as α runs over Z(p). Our main results show that the ratios τ (p,α)/ω (p,α) and τ (p)/ω (p) are bounded above and below by constants that only depend on the degree of p. In particular, we prove that (formula), for any polynomial of degree n

Linear maps between C*-algebras whose adjoints preserve extreme points of the dual ball

We give a structural characterisation of linear operators from one C -algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a -isomorphism, such a map admits of a decomposition into a degenerate and a nondegenerate part, the non-degenerate part of which appears as a Jordan -morphism followed by a "rotation" and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the "rotation" is but the identity. In this context the results concerning such pure state preserving maps depend on and complement those of Størmer [Stø2; 5.6 & 5.7]. In conclusion we consider the action of maps with "extreme point preserving" adjoints on some specific C -algebras. 1 Introduction It is clear from the remarks made in the abstract that the results concerning maps with "pure state preserving" adjoints, provide us with a valuable clue as to what objects we may regard as "non-commuta..