Une généralisation de l'inégalité de Stein-Lorenzini

Let K be a field of an arbitrary characteristic, K¯ an algebraic closure of K and P a nonconstant polynomial of n⩾2 variables, with coefficients in K. For λ∈K¯, denote the number of distinct irreducible factors fλ,i in a factorization of P−λ over K¯ by n(λ). We show the following statement, which generalizes previous results of Stein (1989) and Lorenzini (1993): if P is noncomposite over K¯ then ∑λ(n(λ)−1) is at most equal to minλ{∑ideg(fλ,i)}−1.