Asymptotic Directions for the Zero Sets of the Components of an Electrical Field from a Finite Number of Point Charges on the Plane Part II

We study the structure of the zero set of a nontrivial finite point charge electrical field $F=(X,Y)$ in the plane ${\mathbb{R}}^2$. We establish equations satisfied by the possible directions for the zero sets {X = 0} and $\{Y=0\}$ separately, and we show that there are only finitely many possible asymptotic directions for both of these zero sets. We suspect that the set of asymptotic directions for {X = 0} and the set of asymptotic directions for $\{Y=0\}$ are (essentially) distinct.