arXiv subjects

André Galligo

Publications and source records attributed to André Galligo.

10 recordsLinked to original sources

arXiv2025

Dynamics of rotationally invariant polynomial root sets under iterated differentiations

We associate to an $N$-sample of a given rotationally invariant probability measure $μ_0$ with compact support in the complex plane, a polynomial $P_N$ with roots given by the sample. Then, for $t \in (0,1)$, we consider the empirical measure $μ_t^{N}$ associated to the root set of the $\lfloor t N\rfloor$-th derivative of $P_N$. A question posed by O'Rourke and Steinerberger [21], reformulated as a conjecture by Hoskins and Kabluchko [10], and recently reaffirmed by Campbell, O'Rourke and Renfrew [5], states that under suitable conditions of regularity on $μ_0$, for an i.i.d. sample, $μ_t^{N}$ converges to a rotationally invariant probability measure $μ_t$ when $N$ tends to infinity, and that $(1-t)μ_t$ has a radial density $x \mapsto ψ(x,t)$ satisfying the following partial differential equation: \begin{equation} \label{PDErotational} \frac{ \partial ψ(x,t) }{\partial t} = \frac{ \partial}{\partial x} \left( \frac{ ψ(x,t) }{ \frac{1}{x} \int_0^x ψ(y,t) dy } \right). \end{equation} In [10], this equation is reformulated as an equation on the distribution function $Ψ_t$ of the radial part of $(1-t) μ_t$: \begin{equation} \label{equationPsixtabstract} \frac{\partial Ψ_t (x)}{\partial t} = x \frac{\frac{\partial Ψ_t (x)}{\partial x} } {Ψ_t(x)} - 1. \end{equation} Restricting our study to a specific family of $N$-samplings, we are able to prove a variant of the conjecture above. We also emphasize the important differences between the two-dimensional setting and the one-dimensional setting, illustrated in our Theorem 2.1.

math.PR↗

arXiv2025

Dynamics of roots of randomized derivative polynomials

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature $β\in (0, \infty]$, the case $β= \infty$ corresponding to the situation where one considers the derivative of polynomials, without randomization. Our constructions can be connected to random matrix theory: in particular, as detailed in Section 2, for $β= 2$ and roots on the real line, we get the distribution of the eigenvalues of minors of unitarily invariant random matrices. We prove that the asymptotic macroscopic behavior of the roots, i.e. the hydrodynamic limit, does not depend on $β$, and coincides with what we obtain for the non-randomized iterated derivatives, i.e. for $β= \infty$. Since recent results obtained for iterated derivations show that the limiting dynamics is governed by a non-local and non-linear PDE, we can transfer this information to the macroscopic behavior of the randomized setting. Our proof is completely explicit and relies on the analysis of increments in a triangular bivariate Markov chain.

math.PR↗

arXiv2024

Anti-concentration applied to roots of randomized derivatives of polynomials

Let $(Z^{(n)}_k)_{1 \leq k \leq n}$ be a random set of points and let $μ_n$ be its \emph{empirical measure}: $$μ_n = \frac{1}{n} \sum_{k=1}^n δ_{Z^{(n)}_k}. $$ Let $$P_n(z) := (z - Z^{(n)}_1)\cdots (z - Z^{(n)}_n)\quad \text{and}\quad Q_n (z) := \sum_{k=1}^n γ^{(n)}_k \prod_{1 \leq j \leq n, j \neq k} (z- Z^{(n)}_j), $$ where $(γ^{(n)}_k)_{1 \leq k \leq n}$ are independent, i.i.d. random variables with Gamma distribution of parameter $β/2$, for some fixed $β> 0$. We prove that in the case where $μ_n$ almost surely tends to $μ$ when $n \rightarrow \infty$, the empirical measure of the complex zeros of the \emph{randomized derivative} $Q_n$ also converges almost surely to $μ$ when $n$ tends to infinity. Furthermore, for $k = o(n / \log n)$, we obtain that the zeros of the $k-$th \emph{randomized derivative} of $P_n$ converge to the limiting measure $μ$ in the same sense. We also derive the same conclusion for a variant of the randomized derivative related to the unit circle.

math.PR↗

arXiv2022

Modeling complex root motion of real random polynomials under differentiation

In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment $Δ$t corresponds to 1 n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.

math.NA↗

arXiv2021

Spirometry-based airways disease simulation and recognition using Machine Learning approaches

The purpose of this study is to provide means to physicians for automated and fast recognition of airways diseases. In this work, we mainly focus on measures that can be easily recorded using a spirometer. The signals used in this framework are simulated using the linear bi-compartment model of the lungs. This allows us to simulate ventilation under the hypothesis of ventilation at rest (tidal breathing). By changing the resistive and elastic parameters, data samples are realized simulating healthy, fibrosis and asthma breathing. On this synthetic data, different machine learning models are tested and their performance is assessed. All but the Naive bias classifier show accuracy of at least 99%. This represents a proof of concept that Machine Learning can accurately differentiate diseases based on manufactured spirometry data. This paves the way for further developments on the topic, notably testing the model on real data.

cs.LG↗

arXiv2016

Extraction of cylinders and cones from minimal point sets

We propose new algebraic methods for extracting cylinders and cones from minimal point sets, including oriented points. More precisely, we are interested in computing efficiently cylinders through a set of three points, one of them being oriented, or through a set of five simple points. We are also interested in computing efficiently cones through a set of two oriented points, through a set of four points, one of them being oriented, or through a set of six points. For these different interpolation problems, we give optimal bounds on the number of solutions. Moreover, we describe algebraic methods targeted to solve these problems efficiently.

cs.CG↗

arXiv2016

Algebraic method for constructing singular steady solitary waves: A case study

This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations. The method is illustrated by the study of capillary-gravity steady surface waves propagating in shallow water. We consider the (fully nonlinear, weakly dispersive) Serre-Green-Naghdi equations with surface tension, because it provides a tractable model that, in the same time, is not too simple so the interest of the method can be emphasised. In particular, we analyse a special class of solutions, the solitary waves, which play an important role in many fields of Physics. In capillary-gravity regime, there are two kinds of localised infinitely smooth travelling wave solutions -- solitary waves of elevation and of depression. However, if we allow the solitary waves to have an angular point, the "zoology" of solutions becomes much richer and the main goal of this study is to provide a complete classification of such singular localised solutions using the methods of the effective Algebraic Geometry.

physics.class-ph↗

arXiv2014

Quantitative equidistribution for the solutions of systems of sparse polynomial equations

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f_1=\dots=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials with integer coefficients, and of random systems of Laurent polynomials with complex coefficients.

math.CV↗

arXiv2010

Random polynomials and expected complexity of bisection methods for real solving

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones? We exploit results by Kac, and by Edelman and Kostlan in order to estimate the real root separation of degree $d$ polynomials with i.i.d.\ coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the $i$-th coefficient has variance ${d \choose i}$, whereas for Weyl polynomials its variance is ${1/i!}$. By applying results from statistical physics, we obtain the expected (bit) complexity of \func{sturm} solver, $\sOB(r d^2 τ)$, where $r$ is the number of real roots and $τ$ the maximum coefficient bitsize. Our bounds are two orders of magnitude tighter than the record worst case ones. We also derive an output-sensitive bound in the worst case. The second part of the paper shows that the expected number of real roots of a degree $d$ polynomial in the Bernstein basis is $\sqrt{2d}\pm\OO(1)$, when the coefficients are i.i.d.\ variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis.

cs.SC↗

arXiv2010

Modular Las Vegas Algorithms for Polynomial Absolute Factorization

Let $f(X,Y) \in \ZZ[X,Y]$ be an irreducible polynomial over $\QQ$. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of $f$, or more precisely, of $f$ modulo some prime integer $p$. The same idea of choosing a $p$ satisfying some prescribed properties together with $LLL$ is used to provide a new strategy for absolute factorization of $f(X,Y)$. We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to factorize some polynomials of degree up to 400.

math.AG↗