Enumerating Combinatorial Classes of the Complex Polynomial Vector Fields in the Complex Plane

TL;DR

This paper counts and characterizes the combinatorial classes of complex polynomial vector fields in the plane, extending previous results and providing explicit formulas and asymptotic behavior for these counts.

Contribution

It introduces a bracketing problem approach to enumerate combinatorial classes and derives closed-form and asymptotic formulas for their counts based on degree and dimension.

Findings

Number of combinatorial classes is linked to a bracketing problem.

Derived closed-form expressions for class counts as functions of degree and dimension.

Provided asymptotic analysis for the number of classes as degree increases.

Abstract

In order to understand the parameter space of monic and centered complex polynomial vector fields of degree d in the complex plane, 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 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 of degree d in the complex plane 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 the number of classes, as functions of d and q, and we furthermore make an asymptotic analysis of these sequences for d tending to infinity. These results are also applicable to special classes of Abelian differentials, quadratic differentials with double poles, and singular holomorphic foliations of the plane.