Complex algebraic plane curves via the Poincaré–Hopf formula, I: Parametric lines

Abstract

We present a new and very efficient approach to study topology of algebraic curves in ‫ރ‬ 2 . It relies on using the Poincaré-Hopf formula, applied to a suitable Hamiltonian vector field, to estimate the number of double points and the Milnor numbers of singular points of the curve and on considering finite-dimensional spaces of curves with given asymptotics at infinity. We apply this method to classification of parametric lines with one selfintersection.

Geometry of Puiseux expansions

“…Namely, if c 1 = 0, then a generic curve has 1 2 [(p − 1)(q − 1) − (p − 1)] finite double points. A generic curve with c 1 = 0 has at most 1 2 [(p − 1)(q − 1) − 2(p − 1)] such points, so some double points tend to infinity as c 1 → 0 (see [BZ1]). Lemma 3.3.…”

“…The idea to consider this map has arisen in our investigations of polynomial curves with b 1 = 1 in [BZ1] and of the Jacobian conjecture [Zol]. Definition 1.2.…”

Number of singular points of an annulus in \mathbb{C}^2

“…Another one, due to M. Zaidenberg and V. Lin [11], says that any curve homeomorphic to a disk is algebraically equivalent to a curve of the type x p = y q for p, q coprime. In [3,5] we developed an efficient method in some other particular cases: namely we studied rational curves with one place at infinity and one double point (topological immersions of C in C 2 with one finite self-intersection) in [3] and annuli (topological embeddings of C * in C 2 ) in [5]. A list of 44 possible cases was found and it was claimed that the list is complete.…”

Small amplitude limit cycles for the polynomial Liénard system

“…. are called the essential Puiseux quantities of the singularity X = Y = 0 of the curve C (see [1]). They are related with the Poincaré-Lyapunov quantities g 1 , g 3 , .…”

“…max{ν: c 1 = c 3 = · · · = c 2ν−1 = 0 = c 2ν+1 }; H C (m, n)-the maximal number of limit cycles bifurcating from the origin in the complex sense, i.e. 1 2 × maximal number of zeroes r i = 0 of the function P (r) − r for r ∈ (C, 0) (counted with multiplicities); H * C (m, n)-the maximal cyclicity of x = y = 0 in the complex sense.…”