Jaume Llibre
08193 Bellaterra, Barcelona, Catalonia, Spain
Departament de Matemàtiques, Universitat Autònoma de Barcelona
Publications:
Corbera M., Llibre J., Yuan P.
On the Convex Central Configurations of the Symmetric $(\ell + 2)$body Problem
2020, vol. 25, no. 3, pp. 250272
Abstract
For the $4$body problem there is the following conjecture: Given arbitrary positive masses, the planar $4$body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the $(\ell+2)$body problem with $\ell \geqslant 3$. In particular, we prove that the symmetric $(2n+1)$body problem with masses $m_1=\ldots=m_{2n1}=1$ and $m_{2n}=m_{2n+1}=m$ sufficiently small has at least two classes of convex central configuration when $n=2$, five when $n=3$, and four when $n=4$. We conjecture that the $(2n+1)$body problem has at least $n $ classes of convex central configurations for $n>4$ and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric $(2n+2)$body problem with masses $m_1=\ldots=m_{2n}=1$ and $m_{2n+1}=m_{2n+2}=m$ sufficiently small has at least three classes of convex central configuration when $n=3$, two when $n=4$, and three when $n=5$. We also conjecture that the $(2n+2)$body problem has at least $[(n+1)/2]$ classes of convex central configurations for $n>5$ and we give some numerical evidences that the conjecture can be true.

Llibre J., Valls C.
Darboux Polynomials, Balances and Painlevé Property
2017, vol. 22, no. 5, pp. 543550
Abstract
For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.
