On the minimum value of the condition number of polynomials

2020/12/09 by Beltrán, Carlos, Lizarte, Fátima · 2 citations
#30C15 #30E10 #31A15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.05138

Abstract

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 a previous paper by C. Beltán, U. Etayo, J. Marzo and J. Ortega-Cerdà, 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\geqslant 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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