### Citations

636 |
Singular points in complex hypersurfaces
- Milnor
- 1968
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Citation Context ... index of theseld fW 0 along @Aj : The latter index equals the index of theseld fW j eAj at pj : Summing-up all this over j we get the doubled number of double points of the perturbation. Lemma 2. (=-=[Mil]-=-, [Lins], [BZI]). We have (2.2) 20 = X j 0(Aj) + 2 X i<j (Ai Aj)0; where 0(Aj) is the Milnor numbers of the germ Aj at the point 0 and (Ai Aj)0 is the intersection number at 0 of the components ... |

30 |
An irreducible simply connected algebraic curve in C2 is equivalent to a quasihomogeneous curve
- Lin
- 1983
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Citation Context ...h (1.10). We obtain the following bound (which is weaker than in Main Theorem) Corollary 2. HC(m;n) max: In order to improve this estimate we use the following theorem of M. Zaidenberg and V. Lin =-=[ZaLi]-=-: if an algebraic curve of the form (1.6) has only one singular point then it is equivalent to a quasi-homogeneous curve. Proof of Main Theorem. We know that C is equivalent to a quasi-homogeneous cur... |

20 | On rational cuspidal curves. I. Sharp estimate for degree via multiplicities - Orevkov |

12 |
1999] “Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces,” Nonlinearity 12
- Christopher, Lynch
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Citation Context ...1r(1 + : : :) + g3r3(1 + : : :) + : : : ; r ! 0+; from the section f(x; y) = (r; 0) : r 0)g to itself. Namely, gj are proportional to cj with coe¢ cients depending only on j: We refer the reader to =-=[ChLy]-=- for details. Since thesxed points of the map (1.7) correspond to the limit cycles of the Liénard vectorseld, the essential Puiseux quantities of the curve C become responsible for the small amplitude... |

11 |
Number of zeroes of complete elliptic integrals, Funct
- Petrov
- 1984
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Citation Context ...2 n < m (which agrees with the above) and bHC(n; n) = 2n 4 + 2n (which is stronger than above). Christopher and Lynch proved the formula bH(m; 2) = [ 2m+13 ] = m [m+13 ], using some Petrovs =-=[Pet]-=- ideas. They also proved that bH(m; 3) = 2[ 2(m+2)8 ] when 3 m 50 and bHC(m; 3) = [ 3(m+2)4 ] when 6 m 50: They found examples where bHC(m; 3) > bH(m; 3) (e.g. bHC(7; 3) = 7 and bH(7; 3) = 6):... |

8 |
Principles of Algebraic Geometry
- ths, Harris
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Citation Context ...nly if c1 = c3 = : : : = 0; i.e. (u) = e(u2) is an even function. From the algebraic point of view this means that the curve (1.6) is multiply covered (or non-primitive). By the Lüroth theorem (see =-=[GrHa]-=-) we have F (x) = eF !(x); G(x) = eG !(x) for a polynomial !(x) = x2 + : : :. From the dynamical point of view this means that the system (1.4) is timereversible and the system (1.1) is rationall... |

4 | Żołądek Complex algebraic plane curves via Poincaré–Hopf formula - Borodzik, H |

4 |
Conditions for the center for certain equations of the form yy′
- Cherkas
- 1972
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Citation Context ...ported by Polish KBN Grant No 1 P03A 015 29. 1 2 MACIEJ BORODZIK AND HENRYK ·ZO×ADEK is the Puiseux expansion at the point X = Y = 0 of the curve (1.6) C : X = F (x); Y = G(x): It is well known, see =-=[Che]-=-, that the system (1.1) (equivalently, (1.4)) has center at the origin if and only if c1 = c3 = : : : = 0; i.e. (u) = e(u2) is an even function. From the algebraic point of view this means that the ... |

4 | Orevkov, Some estimates for plane cuspidal curves, in: “Seminaire d’Algébre et Geometrie - Zaidenberg, Yu - 1993 |

2 |
Complex algebraic curves via PoincaréHopf formula
- Borodzik, ·Zo÷adek
(Show Context)
Citation Context ...reversible, i.e. it can be pushed forward via the map (x; y)! (!(x); y): The coe¢ cients c1; c3; c5; : : : are called the essential Puiseux quantities of the singularity X = Y = 0 of the curve C (see =-=[BZI]-=-). They are related with the PoincaréLyapunov quantities g1; g3; : : : ; which appear in the Taylor expansion of the Poincaré return map (1.7) r ! P (r) = r + g1r(1 + : : :) + g3r3(1 + : : :) + : : :... |

2 | Algebraic solutions of polynomial dierential equations and foliations in dimension two, in \Holomorphic dynamics - LinsNeto - 1988 |