We explore the connections between minimizers of the discrete logarithmic energy on the 2-dimensional sphere, univariate polynomials with optimal condition number in the Shub-Smale sense and a quotient involving norms of polynomials.

We prove that polynomials with optimal condition number produce spherical points with small logarithmic energy (the reverse result was proved by M. Shub and S. Smale in 1993) and a sharp Bombieri type inequality for univariate polynomials with complex coefficients.