Autour d'un théorème de Stein

Abstract

Let 𝐾 be a field of characteristic zero, k ̅ an algebraic closure of K and P (X, Y) a non constant polynomial, with coefficients in K. For ¸ λ𝜖 k ̅, denote the number of distinct irreducible factors f⋋,i in a factorization of P– λ ¸ over (k ) ̅by ո(λ). We rewrite without the jacobian derivation aspect of Stein's proof (1989) for showing the following statement : if P is non-composite then ∑ ⋋(n(λ) – 1) is at most equal to deg(P) –1.