Co-circular Deltoid and Isosceles Trapezoid Central Configurations in the 4-body Problem

A central configuration is defined as a system of masses in which the gravitational acceleration vectors point toward the center of mass and are proportional to the displacement vectors from the center of mass with the same proportionality constant. Central configurations play an important role in the study of the Newtonian n-body problem because from these it is possible to construct explicit solutions. We consider two special cases of the 4-body problem. First we study the co-circular deltoid, which is a convex quadrilateral with a line of symmetry going through two opposite vertices. The second case we study are the co-circular isosceles trapezoid configurations. These special cases of the 4-body problem can also be used to construct examples of 5-body configurations. Central configurations can be described using a system of polynomial equations called  the Albouy-Checiner equations. When using techniques from computational algebraic geometry on these equations, it appears that the exponents of the univariate elimination polynomials are multiples of three and the polynomials can be factored into perfect squares. We study the algebraic patterns in the elimination polynomials that occur in these special cases.