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

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