Nilpotent systems admitting an algebraic inverse integrating factor over \(\mathbb{C}((x,y))\)
The authors study the existence of an inverse integrating factors \(V\) for certain planar vector fields. The main reason to study its existence is that, using an inverse integrating factor, it is possible to compute a first integral of the vector field. The inverse integrating factors play an important role in the study of the existence of limit cycles of a vector field. The zero-set \(\{V = 0 \}\) also contains the homoclinic and heteroclinic connections between hyperbolic saddle equilibria. Moreover, the cyclicity of a limit cycle is related with the vanishing order of \(V\). The expressions of \(V\) usually are simpler than the expressions of the first integrals and the domain of definition and the regularity of \(V\) usually are larger than the domain and the regularity of the first integral. The authors characterize the nilpotent systems which have an algebraic inverse integrating factor over \(\mathbb{C}((x, y))\) where \(\mathbb{C}((x, y))\) denotes the quotient field of the algebra of the power series \(\mathbb{C}x, y\). In previous works of the authors, the nilpotent systems with a formal inverse integrating factor were characterized. In the present paper, the authors characterize the nilpotent systems whose lowest-degree quasihomogeneous term is \((y, \sigma x^n)^T\), \(\sigma = \pm 1\), which have an algebraic inverse integrating factor over \(\mathbb{C}((x, y))\). In such cases, it is shown that the systems admit an inverse integrating factor of the form \((h+\dots )^q\) with \(h = 2 \sigma x^{n+1}-(n+1)y^2\) and \(q\) a rational number.
scientific article; zbMATH DE number 6134602
scientific article