The monodromy group at infinity of a generic polynomial vector field on the complex projective plane
The authors study an analytic differential equation with a finite set of singularities on \(\mathbb{C}^2\) in an affine chart, written in the form \[ {dw\over dz}= {P_n\over Q_n},\tag{1} \] where \(P_n\) and \(Q_n\) are polynomials of degree, \(n\). \({\mathcal A}\) denotes the space of complex coefficients of equation (1) and \({\mathcal A}^{\mathbb{R}}\) the space of equations (1) with real coefficients. The authors study an application of the general principle that usually the varieties defined in the space of coefficients by natural conditions imposed on equation (1) are not algebraic. An elegant proof extracted from the previous general principle proves that for a generic polynomial vector field on the complex projective plane the monodromy group at infinity is free. More precisely, the following three theorems are very well proven with a critical development provided for the prerequisite mathematical machinery required for their completions. The theorems are as follows: Theorem 1 (main theorem). For a generic polynomial vector field on \(\mathbb{C}^2\) of degree \(n>2\), the monodromy group at infinity is free with \(n\) generators. We recall that a group with a finite set of generators is free if there are no relations among the generators. This theorem admits the following equivalent form. Theorem \(1'\). All cycles on the line at infinity of a generic rational equation of degree \(n>2\) are limit cycles. Theorem 2. For any polynomial of degree \(n+1\), \(n>2\), and for any complex cycle on the phase curve of the corresponding Hamiltonian equation there exists a perturbation of this cycle or makes it a limit cycle. The proofs are very elegantly and completely proven implementing some very outstanding methods.
scientific article; zbMATH DE number 1206259