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

TL;DR

This paper establishes bounds on the total multiplicity of reducible algebraic plane curves in a pencil, improving previous estimates by incorporating multiplicities and Newton polygons.

Contribution

It introduces a new bound on the total multiplicity of reducible curves in a pencil, considering multiplicities and Newton polygons, which refines earlier results.

Findings

Total multiplicity of reducible curves is bounded by d^2 - 1

Inclusion of multiplicities provides a more accurate count

Newton polygon analysis yields sharper bounds

Abstract

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^2-1 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.