Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

TL;DR

This paper presents a polynomial-time algorithm for computing Darboux polynomials and rational first integrals of planar polynomial differential systems with bounded degree, improving efficiency over traditional methods.

Contribution

It introduces a new polynomial-time algorithm for computing irreducible Darboux polynomials and rational first integrals, surpassing previous exponential-time approaches.

Findings

Lagutinskii-Pereira's algorithm computes Darboux polynomials efficiently

The new method operates in polynomial time relative to system parameters

It can determine the existence of rational first integrals with bounded degree

Abstract

In this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y), dY/dt= B(X, Y), where A,B belongs to Z[X, Y], degA \leq d, degB \leq d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y)dX + B(X, Y)dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira's algorithm computes irreducible Darboux polynomials with degree smaller than N, with a polynomial number, relatively to d, log(H) and N, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree.