Abstract:

In 1998, S. Smale proposed the following problem for mathematicians of this century: "Consider n bodies with positive masses . Is the number of corresponding central configurations finite? "

The question of deciding whether the Smale Conjecture is true is one of the most important problems of central configurations.

When Moeckel and Hamptom have proved that the Smale conjecture is true. When Albouy and Kaloshim proved the conjecture for a generic choice of masses.

In 2017 I published a work where I propose the use of the Jacobian criterion for the obtainment of finiteness results for generic choice of the masses in which I reobtained a generic fineness result for Dziobek Configurations obtained by Moeckel in 2001 that did makes use of resultants. The Jacobian criterion has the advantage of being elementary and relatively inexpensively computationally.

When , the Smale conjecture still open when the central configuration has a symmetry axis. The aim of this talk is to define a algebraic set V that contains every Symmetric Planar Central Configurations of the Five Body Problem and use the Jacobian Crieterion in order to provide a upper bound for the dimension of V. This is a work in progress.