On the Finiteness Issue of Four-Body Balanced Configurations in the Plane

For any given positive masses, we prove that the number of $\mathbf{S}$-balanced configurations of four bodies in the plane is finite up to similitudes, provided that the symmetric matrix $\mathbf{S}$ is sufficiently close to a numerical matrix. To establish this result, we utilize singular sequences to analyze the possible degenerate algebraic varieties defined by $\mathbf{S}$-balanced configurations. We derive all potential singular diagrams, encompassing both equal-order and non-equal-order cases. In the equal-order case, we obtain the necessary mass equations, while for $\mathbf{S}$ approaching the identity matrix, we demonstrate the absence of non-equal-order singular sequences, thereby rigorously rule out all non-generic scenarios. Furthermore, we extend this conclusion to the five-body scenario.