Enumerating combinatorial classes of the complex polynomial vector fields in ℂ

Abstract

In order to understand the parameter space Ξ d of monic and centered complex polynomial vector fields in C of degree d, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity).This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) dimension q in parameter space, for fixed degree d; these numbers are denoted by c d and c d,q respectively. These results are extensions of a result by Douady, Estrada, and Sentenac, which shows that the number of combinatorial classes of the structurally stable complex polynomial vector fields in C of degree d is the Catalan number C d−1 .We show that enumerating the combinatorial classes is equivalent to a so-called bracketing problem. Then we analyze the generating functions and find closed-form expressions for c d and c d,q , and we furthermore make an asymptotic analysis of these sequences for d tending to ∞.These results are also applicable to special classes of Abelian differentials, quadratic differentials with double poles, and singular holomorphic foliations of the plane. Prepared using etds.cls [Version: 1999/07/21 v1.0] 2000 Mathematics Subject Classification: 37F75, 05A15, 05A16.

Topological classification of polynomial complex differential equations with all the critical points of centre type

“…In our context, the only separatrices will be the ones of the saddle points at infinity. The problem of studying the number of different phase portraits of equation (1.1), without the assumption of having all the critical points of center type, is also considered in the recent papers [4,9].…”

Analytic normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold

“…In each of these regions we can draw a curve γ j connecting the interior of a saddle sector at ∞ to the interior of another saddle sector (see Figure 6.3(B)). There are exactly C k = 1 k+1 2k k ways of pairing two by two the saddle sectors of ∞ by non-intersecting curves, thus providing a topological invariant for the vector field (we also refer to [10]).…”

“…The paths of integration Γ j,± bounds the unbounded squid sectors in the following way : The boundary of the saddle part V j,s of (unbounded) squid sectors is Γ j,+ ∪ Γ j+1,− , as in Figures 6. 10 (1) For every (x, y) ∈ V j,s r F j+1 (x, y) − F j (x, y) = 2iπT j H j N (x, y) (6.23) while for every (x, y) ∈ V σ (j),g r F j (x, y) = F σ (j) (x, y) .…”

“…Also, we have refined our understanding of the modulus compared to the presentation in [40]. The number of cells is now the optimal number C k = 1 k+1 2k k (the k th Catalan's number) given by the Douady-Sentenac classification [11,10]. Moreover we have reduced the degrees of freedom: instead of having the modulus given up to conjugacy by linear functions depending both on ε and the cell, now the modulus is given up to conjugacy by linear functions depending only on ε in an analytic way.…”

Analytic Normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold

“…Also, we have refined our understanding of the modulus compared to the presentation in [33]. The number of cells is now the optimal number C k = 1 k+1 2k k (the k th Catalan's number) given by the Douady-Estrada-Sentenac classification [8,7]. Moreover we have reduced the degree of freedom: instead of having the modulus given up to conjugacy by linear functions depending both on ε and the cell, now the modulus is given up to conjugacy by linear functions depending only on ε in an analytic way.…”