From Gauss–Lucas to de Bruijn–Springer

The main result

Dear Reader, in today's brief we leap from Gauss–Lucas to de Bruijn–Springer. For decades mathematicians chased a clean link between the roots of a polynomial and those of its derivative. Gauss–Lucas gives the first half of it: every root of $p'(z)$ sits inside the convex hull of the roots of $p(z)$, so each one can be written as a convex combination of them — the rows of the weight matrix sum to one.

The de Bruijn–Springer conjecture of 1948 went further. The roots of the derivative are not merely inside the hull, they are pulled inward on average, and the slide below says in what sense. The $(n-1) \times n$ matrix of convex weights can be chosen so that its columns sum to $1 - \frac{1}{n}$ as well. Every root of $p$ then carries exactly the same total weight, and the consequence is a majorisation: for every convex function, the average over the critical points does not exceed the average over the roots.

The conjecture stayed open for 55 years. Then, in 2003, Semyon Malamud and Rajesh Pereira independently cracked it, each with a short and elegant argument. Why do breakthroughs so often arrive in pairs? Please enjoy this beautiful result below.