Related papers: A universal method to approach the Poincar\'e cent…

We discuss the local behaviour of vector fields in the plane $\R^2$ around a regular singular point, using recently introduced reduced normal forms, i.e. Poincar\'e and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997),…

In recent years, the so-called polynomial moment problem, motivated by the classical Poincare center-focus problem, was thoroughly studied, and the answers to the main questions have been found. The study of a similar problem for rational…

The Polynomial Abel differential equations are considered a model problem for the classical Poincar\'e center--focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all…

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and…

We consider parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $x\in\mathbb{R}^n_>$ such that ${A \, (c \circ x^B)=0}$ with…

We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets…

The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincar\'e compactification: this…

Let $X$ be a polynomial vector field in $\mathbb{R}^2$ which, after one-point compactification of the plane, has a punctured neighbourhood $\dot U$ of the point at infinity which is foliated by closed orbits of $X$. If the period function…

To study singularities on complex varieties we study Poincar\'e series of filtrations that are defined by discrete valuations on the local ring at the singularity. In all previous papers on this topic one poses restrictions on the centers…

In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincar\'e rank 1 at the origin over a fixed neighborhood $D_r$. The unfolding…

In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these…

We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are…

Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector…

This paper deduces exponential matrix concentration from a Poincar\'e inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof…

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and quartic systems. Using the correction of a…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

We investigate the structure of the centralizer and the normalizer of a local analytic or formal differential system at a nondegenerate stationary point, using the theory of Poincar\'e-Dulac normal forms. Our main results are concerned with…

In this paper we consider the normal map of a closed plane curve as a vector field on the cylinder. We interpret the critical points geometrically and study their Poincar\'{e} index, including the points at infinity. After projecting the…

We show that, to find a Poincare-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that…

We show that any probability measure satisfying a Matrix Poincar\'e inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carr\'e du champ…