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A symmetric planar central configuration of the Newtonian six-body problem$x$ is called cross central configuration if there are precisely four bodies ona symmetry line of $x$. We use complex algebraic geometry and Groebner basistheory to prove that for a generic choice of positive real masses$m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross centralconfigurations. We also show one explicit example of a configuration in thisclass. A part of our approach is based on relaxing the output of the Groebnerbasis computations. This procedure allows us to obtain upper bounds for thedimension of an algebraic variety. We get the same results considering crosscentral configurations of the six-vortex problem.

Generic finiteness for a class of symmetric planar central configurations of the six-body problem and the six-vortex problem