The Casas-Alvero conjecture for three recycled roots in degree 20

The Casas-Alvero conjecture says that a degree $n$ complex univariate polynomial sharing a root with each of its derivative must have only one root. In this article we give three results. The first one, is that the number of possible counterexamples in normal form of degree $p^r+p^s$ or $p^r+2p^s$ is finite ($p$ prime, $r,s$ positive integers). The second result is that a possible counterexample in normal form of degree $p^r+1$ has algebraic coefficients and the final result is that in degree $20$ there are no counterexamples with three recycled roots.