Related papers: The Poincar\'e Problem, algebraic integrability an…
Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where…
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images…
This paper deals with the problem of limit cycle bifurcations for piecewise smooth integrable differential systems with four zones. When the unperturbed system has a family of periodic orbits, the first order Melnikov function is derived…
We study the invariant theory of singular foliations of the projective plane. Our first main result is that a foliation of degree m>1 is not stable only if it has singularities in dimension 1 or contains an isolated singular point with…
This paper revisits an algorithm for isolating real roots of univariate polynomials based on continued fractions. It follows the work of Vincent, Uspen- sky, Collins and Akritas, Johnson and Krandick. We use some tricks, especially a new…
Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In \cite{JCAM}, we have introduced the basis for the present implementation. The particular form of such systems allows reducing…
This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential…
Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e and A. Denjoy. The long-standing problem of generalising these results for the equations of the order $k>1$ (or for the…
In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal')…
In this work we deal with dicritical divisors, curvettes and polynomials. These objects have been one of the main research interests of S.S. Abhyankar during his last years. In this work we provide some elementary proofs of some S.S.…
Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem: given two regular input languages of finite words, decide whether there…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the…
Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, i) for each second-order Lagrangian density on an arbitrary fibred manifold $p\colon E\to N$ the Poincar\'e-Cartan form of which is…
The rigorous solution to the grating diffraction problem is a cornerstone step in many scientific fields and industrial applications ranging from the study of the fundamental properties of metasurfaces to the simulation of photolithography…
The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first…