A dimension bound 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 $n$ is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we produce dimension bounds for this variety (in degree $n$) over any algebraically closed field of characteristic zero using an arithmetic invariant $B(n)$ of $n$. Consequently for infintely many degrees $n$, we show that the associated arithmetic Casas-Alvero scheme has finitely many rational points over any characteristic zero field. Along the way, we prove several rigidity results towards the conjecture and study arithmetic properties of $B(n)$.

arXiv commentUpdated main results and section 5. Sections 2-4 remain unchanged