6533b836fe1ef96bd12a0a58
RESEARCH PRODUCT
Nilpotence of orbits under monodromy and the length of Melnikov functions
Pavao MardesićPavao MardesićJessie Pontigo-herreraDmitry NovikovL. Ortiz-bobadillasubject
PhysicsPure mathematicsSequencePolynomialConjectureMelnikov functionAbelian integrals010102 general mathematicsStatistical and Nonlinear PhysicsIterated integralsCondensed Matter Physics01 natural sciencesNilpotence classFoliationDisplacement functionLimit cyclesMonodromySimple (abstract algebra)[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]Product (mathematics)0103 physical sciences010307 mathematical physics0101 mathematicsOrbit (control theory)ComputingMilieux_MISCELLANEOUSdescription
Abstract Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-12-01 |