Darboux integral

A first integral of a polynomial vector field on the plane, which has a specific form, the product of (non-integer) powers and exponentials of rational functions.

For upper and lower integral sums of a Riemann integrable function, see Darboux sums.

Invariant curves, factors and cofactors

Let $v=P(x,y)\partial_x+Q(x,y)\partial_y$ be a polynomial vector field on the plane, with $P,Q\in\R[x,y]$, having only isolated singularities (i.e., $\gcd(P,Q)=1$); denote by $\omega$ the annulating $V$ polynomial $1$-form, $\omega=-Q(x,y)\rd x+P(x,y)\rd y$: $\omega\cdot v\equiv0$.

An reduced (square-free) algebraic curve $\Gamma=\{R(x,y)=0\}$ is called an invariant curve, or particular integral of the field $v$ (resp., the form $\omega)$, if $v$ is tangent to the curve at all smooth points of the latter. This means that the polynomial $\rd R\cdot v$ vanishes on $\Gamma$, hence is divisible by $R$: $$ \rd R\cdot v=KR,\qquad K=K(x,y)\in\R[x,y]. $$ The polynomial $K$ is called the cofactor of the curve $\Gamma$.

If the polynomial $R$ is reducible, $R=R_1\cdots R_k$ (without repetitions), then each of the irreducible invariant curves $\Gamma_i=\{R_i=0\}$ is a particular integral with the corresponding cofactors $K_i$ defined by the equations $\rd R_i\cdot v = R_i K_i$, $i=1,\dots,k$. Obviously, $K=K_1+\cdots+K_k$.

Computation of the degrees

Let $n=\deg v=\max(\deg P,\deg Q)$ be the degree of the vector field. In general, it is not possible to place an upper bound on the degree $m=\deg \Gamma=\deg R$ of its integral curves, the degree of the cofactor is always $n+(m-1)-m=n-1$. Thus the space of possible cofactors is finite-dimensional and its dimension is $\frac12 n(n+1)$. This number is the natural bound for the number of "nontrivial" invariant curves.

Theorem (G. Darboux). If a polynomial vector field has $k\ge \frac12 n(n+1)+1$ different irreducible algebraic invariant curves $\Gamma_1=\{R_1=0\},\dots\Gamma_k=\{R_k=0\}$, then the field admits a first integral $H$, such that $\rd H\cdot v\equiv0$, of the form $$ H(x,y)=R_1^{\lambda_1}(x,y)\cdots R_k^{\lambda_k}(x,y),\qquad \lambda_1,\dots,\lambda_k\in\C, $$ with the complex exponents $\lambda_1,\dots,\lambda_k$ not all equal to zero. Theorem (J.-P. Jouanolou, 1979). If the number $k$ of the different irreducible algebraic invariant curves is greater or equal to $\frac12 n(n+1)+2$ (one more than before), then $v$ has a rational first integral $$ H(x,y)=\frac{F(x,y)}{G(x,y)},\qquad F,G\in\C[x,y],\ G\not\equiv0. $$