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

TL;DR

This paper investigates the Casas-Alvero conjecture for degree 20 polynomials, proving the nonexistence of counterexamples with three recycled roots and establishing finiteness results for certain degrees.

Contribution

It provides new finiteness results for potential counterexamples and confirms the conjecture holds for degree 20 with three recycled roots.

Findings

No counterexamples with three recycled roots in degree 20

Finiteness of counterexamples in degrees of form p^r+p^s or p^r+2p^s

Counterexamples of degree p^r+1 have algebraic coefficients

Abstract

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.