2025/01/16 by Soham Ghosh, Ghosh, Soham
Mathematics · #12E05 #13D02 #13D03 #14M10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2501.09272
openalex publication_date 2025/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Casas-Alvero conjecture states that if f(X) is a monic univariate polynomial of degree d over a characteristic 0 field \mathbbK such that gcd(f, fi) is non-trivial for each i=1, …, d-1, then f(X)=(X-α)d for some α∈ \mathbbK. In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree d≥ 3 over any characteristic 0 field, by using Koszul homology. Along the way we show existence of various "almost counterexamples" over ℂ, satisfying mildly weaker hypotheses, using Brouwer degree techniques.