Around 2001, motivated by his work on singularities, Casas-Alvero conjectured the following: If a monic univariate polynomial \$f(X)\$ of degree \$n\$, over a field of characteristic \$0\$, has a non-trivial gcd with each of its formal derivatives \$f^{(i)}(X)\$ for \$1\leq i\leq n-1\$, then \$f(X)\$ is a pure power of a monic linear polynomial. In this talk, we will show that over any field of characteristic \$p>0\$, there are "finitely many" counter-examples to the conjecture, and also sketch a proof of the conjecture over characteristic \$0\$. As a corollary, we will also obtain a new proof of the fact that rational normal curves are set-theoretic complete intersections (in characteristic \$0\$). Based on the preprints arxiv: 2402.18717 and arxiv:2501.09272.