Groupes d'automorphismes de $({\bbfC},0)$ et équations différentielles $ydy+\cdots =0$

Cerveau, Dominique; Moussu, Robert

doi:10.24033/bsmf.2108

Cited by 88 publications

(76 citation statements)

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“…The moduli space of F 0 relative to the quasi-homogeneous class is defined by The proof follows from the analytic and topological classifications below using a similar construction as in section 5. We can find the analytic classification in [12] and [3]. Although the original proof was given for α = 2, β = 3 and d = 1 the techniques apply to the general case:…”

Section: Vol 78 (2003)mentioning

confidence: 99%

Moduli spaces of germs of holomorphic foliations in the plane

Marín

2003

Comment. Math. Helv.

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Abstract. In this paper we study the topological moduli space of some germs of singular holomorphic foliations in (C 2 , 0). We obtain a fully characterization for generic foliations whose vanishing order at the origin is two or three. We give a similar description for a certain subspace in the moduli space of generic germs of homogeneous foliations of any vanishing order and also for generic quasi-homogeneous foliations. In all the cases we identify the fundamental group of these spaces using the Gassner representation of the pure braid group and a suitable holonomy representation of the foliation.

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Section: Vol 78 (2003)mentioning

confidence: 99%

Moduli spaces of germs of holomorphic foliations in the plane

Marín

2003

Comment. Math. Helv.

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“…Odani in [3] investigated algebraic phase curves of (1). He proved that if n ≤ m and f m g n (f m /g n ) ≡ 0, (2) then the vector field (1) does not have any algebraic invariant curve.…”

Section: The Resultsmentioning

confidence: 99%

“…. There after suitable resolution of the singularity one obtains an exceptional divisor with singular points of resonant type (see [2]). However, the problem of analytic normal form for such singularities is not solved.…”

Section: Remark It Is Not Clear That the Constants D And Dmentioning

confidence: 99%

Algebraic invariant curves for the Liénard equation

Żołądek

1998

Trans. Amer. Math. Soc.

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Abstract. Odani has shown that if deg g ≤ deg f then after deleting some trivial cases the polynomial systemẋ = y,ẏ = −f (x)y − g(x) does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of deg f and deg g. We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system. The resultsThe subject of this work is the polynomial Liénard systeṁwhereare polynomials of degrees m and n respectively (a m b n = 0). Odani in [3] investigated algebraic phase curves of (1). He proved that if n ≤ m andthen the vector field (1) does not have any algebraic invariant curve.He also presented the example of Wilson [6] withand the algebraic limit cycleOdani conjectured that if m < n < 2m + 1 then (1) does not have algebraic limit cycles. Here we show that this conjecture is false, and we solve the problem of algebraic limit cycles and algebraic invariant curves for all pairs (m, n) almost completely.In what follows we consider systems of the form (1) with the restriction (2) and denote by A m,n the space of such systems,

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“…We fix affine coordinates (x, y) ∈ C 2 such that L ∞ = CP (2) \ C 2 is not invariant, S ∩ L ∞ , Sing(F ) ∩ S ⊂ C 2 . As we said before, the holonomy group H of S \ Sing(F ) is conjugated, according to [5], to a subgroup H ⊆ H p , for some p ∈ N. As a matter of fact, H depends on D. CERVEAU AND P. SAD the transversal section to S as we choose to define it, but we take for granted that H is fixed once and for all.…”

Section: Appendixmentioning

confidence: 99%

“…Then H is holomorphically conjugated to a subgroup of H p for some p ∈ N (see [5]); we say that p ∈ N is the ramification order of H (by the way, we remark that any subgroup of H p is solvable). Let us give two examples.…”

Section: Solvable Holonomy Groups and Foliationsmentioning

confidence: 99%