Darboux integrability (polynomial vector fields)

Darboux integrability is a theory for polynomial vector fields (usually on ${\mathbb{C}}^2$ or ${\mathbb{R}}^2$) that exploits invariant algebraic curves to construct first integrals or integrating factors.

Darboux polynomials

Let $X$ be a polynomial vector field. A polynomial $f\in \mathbb{C}[x,y]$ is a Darboux polynomial (or invariant algebraic curve) of $X$ if

for some polynomial $K$ (the cofactor). Geometrically: the algebraic curve $f=0$ is invariant under the flow of $X$.

Darboux's theorem

If $X$ admits Darboux polynomials $f_1,\dots ,f_k$ with cofactors $K_1,\dots ,K_k$, then:

• If there exist ${\lambda }_i\in \mathbb{C}$ (not all zero) such that $\sum {\lambda }_i{K}_i=0$, the function

  $$F=\prod _{i=1}^k{f}_i^{{\lambda }_i}$$

  is a Darboux first integral of $X$.

• If $\sum {\lambda }_i{K}_i+\mathrm{div}(X)=0$, then $\mu =\prod f_i^{{\lambda }_i}$ is an integrating factor.

A system is Darboux integrable if it admits such a first integral or integrating factor.

Related

• first integral
• integrating factor
• integrable system
• ODE