A SEQUENCE OF POLYNOMIALS WITH OPTIMAL CONDITION NUMBER

An old open problem posed by Mike Shub and Stephen Smale is solved in this paper:

Can one find a sequence of polynomials whose condition number is quasioptimal? The answer is… Yes!

We do so by finding a collection of spherical points that have good quadrature properties and small logarithmic energy. The polynomial whose zeros correspond to these spherical points turn out to be extremelly well conditioned!

Based on a paper by Carlos Beltrán, Ujué Etayo, Jordi Marzo and Joaquim Ortega-Cerdà.

Published in Journal of the American Mathematical Society

DOI:https://doi.org/10.1090/jams/956