2023/07/12 by Daniel Schaub, Schaub, Daniel, Mark Spivakovsky +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · doi:10.48550/arxiv.2307.05997
openalex publication_date 2023/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives Hi(f) is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree d, compile a list of bad primes for that degree (namely, those primes p for which the conjecture fails in degree d and characteristic p) and then deduce the conjecture for all degrees of the form dp^ℓ, ℓ∈ℕ, where p is a good prime for d. In this paper we calculate certain distinguished monomials appearing in the resultant R(f,Hi(f)) and obtain a (non-exhaustive) list of bad primes for every degree d∈ℕ∖\0\.