On pairs of polynomial planar foliations (Q2475544)

The paper deals with topological aspects of pairs of foliations in the plane represented by polynomial 1-forms. A 1-form \(\alpha\) is said to belong to the set \(H_{m}\) if it is a planar homogeneous differential form of degree \(m.\) A pair in similar situation is said to belong to \(A^{k}=H_{m}\times H_{n},\) where \(k=m+n.\) Two pairs \(\left( \alpha_{1},\beta_{1}\right) \) and \(\left( \alpha_{2},\beta_{2}\right) \) in \(A^{k}\) are called equivalent if there exists a homeomorphism \(h\) establishing simultaneous equivalence between the pair of foliations. The authors give a complete characterization of the structural stability in \(A^{k}.\) First they study pairs of planar homogeneous foci and exhibit a topological invariant for the structural stability. Then necessary and sufficient conditions for structural stability of such pairs are derived . Both global and local aspects are considered. The same problems are addressed for the case where \(\alpha\) and/or \(\beta\) are not foci. Finally, conditions for a planar polynomial pair of 1-forms to be finite are presented.