The Zero Set of an Electric Field from a Finite Number of Point Charges

Abstract

We consider the structure of the zero set in ${\mathbb R}^3$ of the electric vector field ${\mathbf F}=(X,Y,Z)$ from a finite set of point charges. We are most interested in the case where the point charges all lie in a plane, and we consider just the zero set in ${\mathbb R}^2$ of the electric vector field ${\mathbf F}=(X,Y)$ from the finite set of point charges. The conjecture is that the zero set of ${\mathbf F}=(X,Y)$ in ${\mathbb R}^2$ is finite. We show fairly easily that this conjecture is true in a Special Case: when the point charges for ${\mathbf F}=(X,Y)$ lie on a line, and we consider the possible zeros throughout ${\mathbb R}^2$. However, even in this Special Case, it is hard to get complete structural information about the zero sets of $X$ and $Y$ separately. We describe structural information about the asymptotic directions at infinity of these two zero sets, and relate this to the interlacing of the zero sets of sequences of polynomials. Then we consider the General Case where the point charges can be anywhere in the plane. As in the Special Case, we construct sequences of polynomials whose zero sets include the asymptotic directions at infinity of the zero sets of $X$ and $Y$ separately. But now the asymptotic directions are not necessarily interlacing, and the structure of the zero sets of these polynomials is less evident. Nonetheless, using these polynomials it might be possible to show that the zeros of ${\mathbf F}=(X,Y)$ are bounded, and then perhaps also, as a result, confirm the conjecture that the zero set of ${\mathbf F}=(X,Y)$ is finite.