On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d − 1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton’s polygon of the pencil. Introduction Given a pencil of algebraic plane curves such that a general element is irreducible, the purpose of this paper is to give a sharp upper bound for the number of reducible curves in this pencil. This question has been widely studied in the literature, but never, as far as we know, by counting the reducible factors with their multiplicities. Let r(X,Y ) = f(X,Y )/g(X,Y ) be a rational function in K(X,Y ), where K is an algebraically closed field. It is commonly said to be non-composite if it cannot be written r = u ◦ h where h(X,Y ) ∈ K(X,Y ) and u ∈ K(T ) such that deg(u) ≥ 2 (recall that the degree of a rational function is the maximum of the degrees of its numerator and denominator after reduction). If d = max(deg(f), deg(g)), we define f (X,Y, Z) = Zf ( X Z , Y Z ) , g(X,Y, Z) = Zg ( X Z , Y Z ) that are two homogeneous polynomials of the same degree d in K[X,Y, Z]. The set σ(f, g) = {(μ : λ) ∈ P K | μf ♯ + λg is reducible in K[X,Y, Z]} ⊂ P K is the spectrum of r and a classical theorem of Bertini and Krull implies that it is finite if r is non-composite. Actually, σ(f, g) is finite if and only if r is non-composite and if and only if the pencil of projective algebraic plane curves μf ♯ + λg = 0, (μ : λ) ∈ P K , has an irreducible general element (see for instance [Jou79, Chapitre 2, Théorème 3.4.6] and [Bod08, Theorem 2.2] for detailed proofs). Notice that the study of σ(f, g) is trivial if d = 1. Therefore, throughout this paper we will always assume that d ≥ 2. Given (μ : λ) ∈ σ(f, g), a complete factorization of the polynomial μf ♯ + λg is of the form (⋆) μf ♯ + λg = n(μ:λ)