A finiteness result towards the Casas-Alvero Conjecture

Abstract

The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The conjecture for polynomials of a fixed degree is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we show that for any algebraically closed field of arbitrary characteristic, this variety is at most two-dimensional for all positive degrees. Consequently, we show that the associated arithmetic Casas-Alvero scheme in any positive degree has finitely many rational points over any field. Along the way, we prove several rigidity results towards the conjecture. We also introduce intermediate arithmetic Casas-Alvero schemes and show that their $\mathbb{K}$ points form an almost complete intersection over any algebraically closed field $\mathbb{K}$. Furthermore, we consider the question of when they form a complete intersection.