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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)

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)

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)

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)

of zeros forcing P to have likewise a bottom row of zeros, and this contradicts the invertibility of P. This argument shows at once that (i) a matrix is invertible if and only if its reduced row echelon form is the identity matrix, and (ii) the set of invertible matrices is precisely the set of products of elementary matrices. Consider the real vector space… (More)

- VANIA MASCIONI, Vania Maschi
- 2016

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- VANIA MASCIONI
- 2012

We prove a tight inequality of non-exponential type for a weighted geometric mean commonly appearing when using Stirling's approximation (also frequently studied in its logarithmic form when computing entropies). As an application we prove corollaries involving binomial coefficients.

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.

- 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.