TY - JOUR
T1 - Inverse problems for multiple invariant curves
AU - Christopher, Colin
AU - Llibre, Jaume
AU - Pantazi, Chara
AU - Walcher, Sebastian
PY - 2007/12/14
Y1 - 2007/12/14
N2 - 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.
AB - 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.
U2 - 10.1017/S0308210506000400
DO - 10.1017/S0308210506000400
M3 - Article
VL - 137
SP - 1197
EP - 1226
JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
SN - 0308-2105
IS - 6
ER -