On the minimum value of the condition number of polynomials

Abstract

Abstract The condition number of a polynomial is a natural measure of the sensitivity of the roots under small perturbations of the polynomial coefficients. In 1993 Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In Beltrán et al. (2021, A sequence of polynomials with optimal condition number. J. Amer. Math. Soc., 34, 219–244) it was proved that the optimal value of the condition number is of the form $\mathcal{O}(\sqrt {N})$, and the sequence demanded by Shub and Smale was described by a closed formula for large enough $N\geqslant N_0$ with $N_0$ unknown, and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $\mathcal{O}(\sqrt {N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer.