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Global properties of a class of planar vector fields of infinite type. (English) Zbl 0882.35029
Consider a 1-form \(\omega= A(x,y)dx+ B(x,y)dy\) in \(\mathbb{R}^2\) with no singularities, that is, \(\omega(p)\neq 0\) for every \(p\). We assume that \(\omega\) is rotationally invariant. The orthogonal to \(\omega\) is generated by the vector field \[ L=- B(x,y) {\partial\over\partial x}+ A(x,y) {\partial\over\partial y}. \] We say \(L\) (or \(\omega\)) has a first integral in an open set \(\Omega\) if there is a differentiable function \(Z:\Omega\to\mathbb{C}\) such that \(LZ=0\) in \(\Omega\) and \(dZ(p)\neq 0\) for every \(p\) in \(\Omega\). The existence of such functions is crucial for the study of local and global properties of the vector field \(L\) such as solvability, regularity, unique continuation, Liouville’s property, etc. When \(\omega\) is assumed to be \(C^\infty\) and complex-valued, we showed in [Manuscr. Math. 89, 355-371 (1996; Zbl 0858.35021)] that it has a global first integral in \(\mathbb{R}^2\) whenever it is of finite type. The first integral was then used to describe several properties of \(L\).
As a natural sequel, here we study \(\mathbb{C}\)-valued, real analytic forms \(\omega\) that are of infinite type along certain of its characteristic circles. In polar coordinates, such a structure is uniquely written as \(M(r)dr+ id\theta\), where \(M\) is a meromorphic function of the variable \(r\). The structure is of infinite type along the circle with radius \(\rho\) if \(M(\rho)=\infty\). It is shown that the residue of \(M\) at \(\rho\) is an invariant of the structure generated by \(\omega\) and that it plays a fundamental role in the integrability and solvability questions.

MSC:
35F05 Linear first-order PDEs
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