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 -