Learn More
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,(More)
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 |α − v| : v ∈ Z(p) \ {α} and τ (p, α) := min |α − v| : v ∈ Z(p) \ {α}. 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(More)
Let Pn be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators T : Pn → Pn for which there exists a constant C > 0 such that for all nonconstant f ∈ Pn there exist a root u of f and a root v of T f with |u − v| ≤ C. We prove that such perturbations leave the degree unchanged and, for a(More)
We confirm two recent conjectures of W. Janous and thereby state the best possible form of the Erdös-Debrunner inequality for triangles. thus the area of the central triangle, while the other three are the areas of the " corner " triangles. The Erdös-Debrunner inequality states that at least one of the corner triangles has no greater area than the central(More)
In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1,. .. , n}, the number D n is defined as the least number of points the space must contain in order to(More)