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- Branko´curgus And, Vania Mascioni
- 2004

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hull of the roots of p, called the Lucas polygon of p. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas… (More)

- Brankó Curgus, Vania Mascioni
- 2005

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)

- Branko´curgus And, Vania Mascioni
- 2008

Let T = α0I + α1D + · · · + αnD n , where D is the differentiation operator and α0 = 0, and let f be a square-free polynomial with large minimum root separation. We prove that the roots of T f are close to the roots of f translated by −α1/α0.

- Branko´curgus And, Vania Mascioni
- 2005

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)

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)

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