Polynomial differential systems with invariant algebraic curves of arbitrary degree formed by Legendre polynomials

Abstract

In 1891 Poincaré asked: Given $m\ge 2$, is there a positive integer $M\left(m\right)$ such that if a polynomial differential system of degree m has an invariant algebraic curve of degree $\ge M\left(m\right)$, then it has a rational first integral? Brunella and Mendes repeated the same open question in 2000, and Lins-Neto in 2002. Between the years 2001 and 2003 three different families of quadratic polynomial differential systems provided a negative answer to this question. One of the answers used the Hermite polynomials. Recently a new negative answer was provided for polynomial differential systems of arbitrary degree using the Laguerre polynomials.

In this paper we provide another new negative answer but using for first time the Legendre polynomials. So the orthogonal polynomials play a role in the Poincaré's question. Moreover we classify the phase portraits of these polynomial differential systems having invariant algebraic curves of arbitrary degree based on the Legendre polynomials.