Constraints on hypothetical counterexamples to the Casas-Alvero conjecture

TL;DR

This paper investigates the Casas-Alvero conjecture, establishing constraints on potential counterexamples, especially focusing on degree 12 and degrees p+1 where p is prime, without fully resolving the conjecture.

Contribution

It provides new necessary conditions that hypothetical counterexamples must meet, narrowing the search for potential counterexamples in specific degrees.

Findings

Counterexamples must have at least 5 distinct roots

Degree 12 remains the first unresolved case

Additional constraints on counterexamples in degree p+1 for prime p

Abstract

The Casas-Alvero conjecture states: if a complex univariate polynomial has a common root with each of its derivatives, then it has a unique root. We show that hypothetical counterexamples must have at least 5 different roots. The first case where the conjecture is not known is in degree 12. We study the case of degree 12, and more generally degree p+1, where p is a prime number. While we don't come closing to solving the conjecture in degree 12, we present several further constraints that counterexamples would have to satisfy.