Weighted Fekete points on the real line and the unit circle

Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to find explicitly, and are only known in a few (unweighted) classical cases. We give two rare explicit examples of weighted Fekete points. The first one is for the weights $w(x)=|x-ai|^{-s}$ on the real line, with $s\ge 1$ and $a\neq 0,$ while the second is for the weights $w(z)=1/|z-b|$ on the unit circle, with $b\in\mathbb{R},\ b\neq\pm 1.$ In both cases, we provide solutions of the continuous energy problems with external fields that express the limit versions of considered weighted Fekete points problems.