Problems around Polynomials - the Good , the Bad and the Ugly

Some very crude estimate on the maximal number of points of equilibrium is obtained in [GNSh] using Khovanskii’s fewnomial theory, see [Kh]. Conjecture 1 is not even settled for 3 positive point charges in which case all points of equilibrium lie in the plane spanned by these charges. (Some special cases of 3 charges are settled in the recent literature, see [Ki],[Pe],[Wa].) Observe that for charges of different signs in R, the set of points of equilibrium might be a space curve. The simplest example of that kind is a 4-tuple of charges placed at the vertices of a square in the z-plane with coordinates (±1,±1, 0). If one places the unit positive charges at (1, 1, 0) and (−1,−1, 0) and the unit negative charges at 2 remaining corners, then the set of points of equilibrium coincides with the z-axis. The following naive-looking question is not settled either.