A class of polynomial planar vector fields with polynomial first integral

TL;DR

This paper presents an algorithm to determine if certain polynomial planar vector fields possess a polynomial first integral and computes the minimal such integral, addressing a specific class of systems with geometric and algebraic significance.

Contribution

The paper introduces a novel algorithm for identifying and computing minimal polynomial first integrals for a class of planar polynomial vector fields with specific geometric properties.

Findings

Algorithm successfully determines existence of polynomial first integrals.

Computes minimal first integrals for systems with curves having one place at infinity.

Solves the Poincaré problem for this class by linking degree computation to singularity reduction.

Abstract

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm computes a minimal first integral. In addition, we solve the Poincar\'e problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field.