Harmonic Properties of the Logarithmic Potential and the Computability of Elliptic Fekete Points

We investigate the properties of the function sending each N-tuple of points to minus the logarithm of the product of their mutual distances. We prove that, as a function defined on the product of N spheres, this function is subharmonic, and indeed its (Riemannian) Laplacian is constant. We also prove a mean value equality and an upper bound on the derivative of the function. We use these results to get sharp upper bounds for the precision needed to describe an approximation to elliptic Fekete points (in the sense demanded by Smale’s 7th problem). We also conclude that Smale’s 7th problem has solutions given by rational spherical points of bounded (small) bit length, proving that there exists an exponential running time algorithm which solves it on the Turing machine model.