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Diego Armentano

Publications and source records attributed to Diego Armentano.

At least 19 recordsLinked to original sources
arXiv2026

Characterization of Logarithmic Fekete Critical Configurations of at Most Six Points in All Dimensions

We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in $\mathbb{R}^d$, in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension $d$. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach. Furthermore, for seven points in $S^2$, we characterize the global minimizer among critical configurations having at least one pair of antipodal points, and give numerical evidence to support the conjecture that this configuration is also the unrestricted global minimizer.

math.AC↗
arXiv2025

On the logarithmic energy of solutions to the polynomial eigenvalue problem

In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two processes are the two extremal particular cases of the polynomial eigenvalue problem, and we prove that the logarithmic energy lies between these two cases. In particular, the roots of the Shub-Smale polynomials are the ones with the lowest logarithmic energy of the family.

math.PR↗
arXiv2025

On a general Kac-Rice formula for the measure of a level set

Let $X(\cdot) $ be a random field $\mathbb{R}^D \to \mathbb{R}^d$, $D\geq d$. We first studied the level set $X^{-1}( u) $, $u \in \mathbb{R}^d$. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the $D-d$ Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise.

math.PR↗
arXiv2023

High Order Numerical Methods To Approximate The Singular Value Decomposition

In this paper, we present a class of high order methods to approximate the singular value decomposition of a given complex matrix (SVD). To the best of our knowledge, only methods up to order three appear in the the literature. A first part is dedicated to defline and analyse this class of method in the regular case, i.e., when the singular values are pairwise distinct. The construction is based on a perturbation analysis of a suitable system of associated to the SVD (SVD system). More precisely, for an integer $p$ be given, we define a sequence which converges with an order $p + 1$ towards the left-right singular vectors and the singular values if the initial approximation of the SVD system satisfies a condition which depends on three quantities : the norm of initial approximation of the SVD system, the greatest singular value and the greatest inverse of the modulus of the difference between the singular values. From a numerical computational point of view, this furnishes a very efficient simple test to prove and certifiy the existence of a SVD in neighborhood of the initial approximation. We generalize these result in the case of clusters of singular values. We show also how to use the result of regular case to detect the clusters of singular values and to define a notion of deflation of the SVD. Moreover numerical experiments confirm the theoretical results.

math.NA↗
arXiv2022

Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$

We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

math.DS↗
arXiv2021

Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

We state the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system. This paper is a continuation of {\it Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems} by the same authors in which the square case was considered. Our main tools are the Kac-Rice formula for the second moment of the volume of the zero set and an expansion of this random variable into the Itô-Wiener Chaos.

math.PR↗
arXiv2020

On the finiteness of the moments of the measure of level sets of random fields

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

math.PR↗
arXiv2018

Conditions for the finiteness of the moments of the volume of level sets

Let X(t) be a Gaussian random field R d $\rightarrow$ R. Using the notion of (d -- 1)-integral geometric measures, we establish a relation between (a) the volume of the level set (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.

math.PR↗
arXiv2018

Central Limit Theorem for the number of real roots of Kostlan Shub Smale random polynomial systems

We obtain a Central Limit Theorem for the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size as the degree goes to infinity. A study of the asymptotic variance of the number of roots is needed, this result was obtained in [2]. Afterwards we represent the number of roots as an explicit non linear functional belonging to the It{ô}-Wiener chaos. This representation provides a tool for applying the Fourth Moment Theorem and henceforth the asymptotic normality. MSC 2010 subject classifications: Primary 60F05, 30C15, ; secondary 60G60, 65H10.

math.PR↗
arXiv2015

Condition length and complexity for the solution of polynomial systems

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale 2000). The main progress on Smale's problem is Beltrán-Pardo (2011) and Bürgisser-Cucker (2010). In this paper we will improve on both approaches and we prove an important intermediate result. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán-Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser-Cucker (2010), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last Theorem is the main result of Bürgisser-Cucker (2010). We give a proof of it relying only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser-Cucker (2010). We build on methods developed in Armentano et al. (2015).

math.NA↗
arXiv2015

A stable, polynomial-time algorithm for the eigenpair problem

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.

math.NA↗
arXiv2014

Smale's Fundamental Theorem of Algebra reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton's method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale's upper bound estimate was infinite average cost. Our's is polynomial in the Bézout number and the dimension of the input. Hence polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser-Cucker where the max of the degrees is greater than or equal to $n^{1+ε}$ for some fixed $ε$. It is possible that Smale's algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.

math.NA↗
arXiv2013

Complexity of Path-Following Methods for the Eigenvalue Problem

A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map appropriate for this context is defined, and a version of Smale's $γ$-Theorem is proven. The main result of this paper bounds the complexity of path-following methods in terms of the length of the path in the condition metric.

math.NA↗
arXiv2010

A review of some recent results on random polynomials over R and over C

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.

math.PR↗
arXiv2009

Random systems of polynomial equations. The expected number of roots under smooth analysis

We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") and the $X_i$'s (the "noise") are independent real-valued Gaussian centered random polynomial fields defined on $\mathbb{R}^m$, 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 infinity. In particular, this means that the behavior of this expected value is governed by the noise part.

math.PR↗