Publicaciones
Publicaciones de Diego Armentano
Armentano, D (2009).
Stochastic Perturbations and Smooth Condition Numbers
Journal of Complexity, 10.1016/j.jco.2010.01.003.
En este artículo definimos un nuevo número de condición adaptado a perturbaciones uniformes en todas las direcciones en un contexto general de mapas entre variedades Riemannianas. Las definiciones y teoremas se pueden aplicar a una gran cantidad de ejemplos. Estudiaremos la relación con el número de condición clásico, y estudiaremos algunos ejemplos importantes.
Armentano, D and Dedieu, J (2009).
A note about the average number of real roots of a Bernstein polynomial system
Journal of Complexity, 25(4):339-342.
We prove that the real roots of normal random homogeneous polynomial systems with n+1 variables and given degrees are, in some sense, equidistributed in the projective space P(R^{n+1}). From this fact we compute the average number of real roots of normal random polynomial systems given in the Bernstein basis.
Armentano, D and Wschebor, M (2009).
Random systems of polynomial equations. The expected number of roots under smooth analysis.
Bernoulli(1):249-266.
We consider random systems of equations over the reals, with m equations and m unknowns P_i(t) + X_i(t) = 0, t 2 Rm, i = 1...m, where the P_i's are non-random polynomials having degrees d_i's (the "signal") and the X_i's (the "noise") are indepen- dent real-valued Gaussian centered random polynomial fields defined on R^m, with a probability law satisfying some invariance properties. For each i, P_ i and X_ i have degree d_ i. The problem is the behavior of the number of roots for large m. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e. P_i identically zero for all i = 1...m), tends to zero geometrically fast as m tends to in¯nity. In particular, this means that the behavior of this expected value, is governed by the noise part.
Armentano, D, Beltrán, C, and Shub, M (2009).
Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials
Transactions of the American Mathematical Society, 363(6):2955-2965.
We prove that points in the sphere associated with roots of random polynomials via the stereographic projection, are surprisignly well-suited with respect to the minimal logarithmic energy on the sphere. That is, roots of random polynomials provide a fairly good approximation to Elliptic Fekete points.
Armentano, D (2010).
A review of some recent results on Random Polynomials over R and over C
(sometido a PMU)..
This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sphere (via the stereographic projection) are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere.
Mario Wschebor: In memoriam
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