Proof of the Casas-Alvero conjecture

The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd (f,f_i)$ is non-trivial for each $i=1,\dots ,d-1$, then $f(X)=(X-\alpha {)}^d$ for some $\alpha \in \mathbb{K}$. In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree $d\ge 3$ over any characteristic $0$ field, by using Koszul homology. Along the way we show existence of various “almost counterexamples” over $\mathbb{C}$, satisfying mildly weaker hypotheses, using Brouwer degree techniques.