Darboux theory of integrability in the sparse case

Darboux’s theorem and Jouanolou’s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.

💡 Research Summary

The paper revisits two classical results in the qualitative theory of polynomial differential equations – Darboux’s theorem on the existence of algebraic first integrals and Jouanolou’s theorem on rational first integrals – and reformulates them in a way that is sensitive to the sparsity of the vector field. In the traditional setting, the bounds guaranteeing the existence of a first integral are expressed solely in terms of the total degree (d) of the polynomial vector field. Such degree‑based bounds are often far from optimal when the vector field is sparse, i.e., when many monomials are missing. The authors propose to replace the degree by a geometric invariant: the size of a Newton polytope associated with the vector field.

A Newton polytope (N) is the convex hull of the exponent vectors of all monomials that appear with non‑zero coefficient in the components of the vector field. The number of integer lattice points (|N\cap\mathbb Z^2|) inside this polytope measures the “effective” combinatorial complexity of the system. The main results can be summarized as follows.

1. Sparse Darboux Theorem (Polynomial First Integral).
   Let (X) be a planar polynomial vector field whose three components share a common Newton polytope (N). Suppose that (X) admits (k) distinct invariant algebraic curves whose defining polynomials have exponent vectors contained in (N). If
   \