2023/12/14 by Daniel Schaub, Schaub, Daniel, Mark Spivakovsky +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2312.08742
openalex publication_date 2023/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Casas--Alvero conjecture predicts that every univariate polynomial f over a field K of characteristic zero having a common factor with each of its derivatives H_i(f) is a power of a linear polynomial. Let f=xd+a_1xd-1+⋯+a_1x ∈ K[a_1,…,a_d-1][x] and let R_i = Res(f,H_i(f))∈ K[a_1,…,a_d-1] be the resultant of f and H_i(f), i ∈ \1,…,d-1\. The Casas-Alvero Conjecture is equivalent to saying that R_1,…,R_d-1 are ``independent'' in a certain sense, namely that the height ht(R_1,…,R_d-1)=d-1 in K[a_1,…,a_d-1]. In this paper we prove a very partial result in this direction : if i ∈ \d-3,d-2,d-1\ then R_i ∉ √(R_1,…,\breveR_i,…,R_d-1.