Algebraic Multiplicity and the Poincar\'{e} Problem

TL;DR

This paper establishes an upper bound for the degree of invariant algebraic curves in polynomial systems using algebraic multiplicities at singular points, introducing a Newton polygon method for computation.

Contribution

It provides a new upper bound for invariant algebraic curves and a novel method to compute algebraic multiplicities via Newton polygons.

Findings

Derived an upper bound for the degree of invariant algebraic curves.

Presented a Newton polygon-based method for algebraic multiplicity calculation.

Applied the results to polynomial systems in the complex projective plane.

Abstract

In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the system at the singular points. A method for computing the algebraic multiplicity using Newton polygon is also presented.