An explicit construction of polynomials with optimal condition numbers

You need to know: Basic complex analysis, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex number z, polynomials in complex variable, root of a polynomial, derivative $P'(z)$ of polynomial $P(z)$ , binomial coefficients ${N\choose i}=\frac{N!}{i!(N-k)!}$ .

Background: For polynomial $P(z)=\sum_{i=0}^N a_i z^i$ with complex coefficients $a_i$ , its Weil norm is $||P|| = \sqrt{\sum_{i=0}^N {N\choose i}^{-1}|a_i|^2}$ , and its (normalised) condition number is $\mu(P) = \max\limits_{z \in {\mathbb C}: P(z)=0} \left(\sqrt{N}\frac{||P||(1+|z|^2)^{N/2-1}}{|P'(z)|}\right)$ .

The Theorem: On 4th March 2019, Carlos Beltrán, Ujué Etayo, Jordi Marzo, and Joaquim Ortega-Cerdà submitted to arxiv a paper in which they proved the existence a constant C and an explicit formula, which, given any integer $N>0$ , produces a polynomial $P_N$ of degree N such that $\mu(P_N) \leq C \sqrt{N}$ .

Short context: A fundamental problem in mathematics is numerical computation of roots of polynomials. Because in numerical studies the coefficients of polynomials are known only approximately, the central issue is to understand what effect on the roots has a slight perturbation of the coefficients. In 1993, Shub and Smale introduced the condition number $\mu(P)$ and observed that for polynomials with small $\mu(P)$ the root computation is stable with respect to the coefficients perturbations. They also proved that a random polynomial P of degree N has $\mu(P)\leq N$ with probability at least $1/2$ , and posed the problem to construct an example of such polynomials explicitly. The Theorem solves this problem. The bound $\mu(P_N) \leq C \sqrt{N}$ is the best possible up to a constant factor.