On the minimum value of the condition number of polynomials

TL;DR

This paper investigates the minimal possible condition number of univariate polynomials of degree N, providing concrete bounds and explicit sequences with condition numbers at most N, advancing understanding of polynomial stability.

Contribution

It offers explicit bounds and a simple formula for polynomial sequences with condition numbers at most N, refining previous asymptotic results.

Findings

Condition number of polynomials is bounded by O(√N).

Explicit sequence with condition number ≤ N for N=4M².

Concrete estimates for the constant in the asymptotic bound.

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. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, 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 $O(\sqrt{N})$ 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.