Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
Stability Of Roots Of Polynomials Under Linear Combinations Of Derivatives, Branko Ćurgus, Vania Mascioni
Stability Of Roots Of Polynomials Under Linear Combinations Of Derivatives, Branko Ćurgus, Vania Mascioni
Mathematics Faculty Publications
Let T=α 0 I+α 1 D+⋅⋅⋅+α n D 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 Tf are close to the roots of f translated by −α 1/α 0.
Perturbations Of Roots Under Linear Transformations Of Polynomials, Branko Ćurgus, Vania Mascioni
Perturbations Of Roots Under Linear Transformations Of Polynomials, Branko Ćurgus, Vania Mascioni
Mathematics Faculty Publications
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 Tf with |u−v|≤C. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of f and Tf, the roots are never displaced by more than a uniform constant independent on f. We show that such "good" operators T …
A Contraction Of The Lucas Polygon, Branko Ćurgus
A Contraction Of The Lucas Polygon, Branko Ćurgus
Mathematics Faculty Publications
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 polygon of p.
On The Location Of Critical Points Of Polynomials, Branko Ćurgus, Vania Mascioni
On The Location Of Critical Points Of Polynomials, Branko Ćurgus, Vania Mascioni
Mathematics Faculty Publications
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, …