Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals axe of special interest. We solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular, we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore, we construct examples in which the genericity assumption does not hold and indicate that the situation is different for these. © 2007 The Royal Society of Edinburgh.
|Journal||Proceedings of the Royal Society of Edinburgh Section A: Mathematics|
|Publication status||Published - 14 Dec 2007|
Christopher, C., Llibre, J., Pantazi, C., & Walcher, S. (2007). Inverse problems for multiple invariant curves. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 137(6), 1197-1226. https://doi.org/10.1017/S0308210506000400