Polynomial differential systems having a given Darbouxian first integral
A function \(H(x,y)=f_1^{\lambda_1}\cdots f_p^{\lambda_p} \exp(g/(f_1^{n_1}\cdots f_p^{n_p}) ),\) where \(f_1,\ldots,f_p\) are irreducible polynomials in \({\mathbb C}[x,y],\) \(\lambda_1,\ldots,\lambda_p\in\mathbb{C},\) \(n_1,\ldots,n_p\in{\mathbb N}\cup\{0\}\) and the polynomial \(g\in {\mathbb C}[x,y]\) is coprime with each \(f_i,\) whenever \(n_i\neq0,\) is called a Darbouxian function. The Darbouxian theory of integrability allows one to determine when a polynomial vector field has a first integral or an integrating factor of Darbouxian type. This paper mainly deals with the inverse problem, namely to characterize the polynomial vector fields \(X=(P,Q)\) in \({\mathbb C}^2\) which admit a first integral of Darbouxian type. The main result is an explicit expression of \(X\) which depends on the fact that \(\max\left(\deg g, \deg(f_1^{n_1}\cdots f_p^{n_p})\right)+\sum_{i=1}^p\deg{f_i}\) is greater or equal \(\deg(X)+1.\) One of the ingredients of the Darboux theory of integrability is the existence of a suitable number of invariant algebraic curves. The paper also deals with the search of Darbouxian integrating factors and gives a result which allows one to ensure the existence of such factors taking into account, among other things, not only the number of invariant algebraic curves, but also their degrees.
scientific article; zbMATH DE number 2132853