In this paper 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. Our main results are that polynomials with optimal condition number produce spherical points with small logarithmic energy (the reverse result was already known) and a sharp Bombieri type inequality for univariate polynomials with complex coefficients.
Problems around polynomials - the good, the bad and the ugly ...
Univariate polynomials and the contractibility of certain sets
A lower bound for the logarithmic energy on $\mathbb{S}^2$ and for the Green energy on $\mathbb{S}^n$
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