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 NN with condition number bounded above by NN?. 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 O(N???)O(N)?, and the sequence demanded by Shub and Smale was described by a closed formula for large enough N?N0N?N0 with N0N0 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 O(N???)O(N) term and we describe a simple formula for a sequence of polynomials whose condition number is at most NN?, valid for all N=4M2N=4M2?, with MM a positive integer.

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