2010/07/28 by Kealey Dias, Dias, Kealey
Mathematics · #05A15 #05A16 #37F75 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1007.5003
openalex publication_date 2010/07/28 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28
In order to understand the parameter space of monic and centered complex\npolynomial vector fields of degree d in the complex plane, decomposed by the\ncombinatorial classes of the vector fields, it is interesting to know the\nnumber of loci in parameter space consisting of vector fields with the same\ncombinatorial data (corresponding to topological classification with fixed\nseparatrices at infinity).\n This paper answers questions posed by Adam L. Epstein and Tan Lei about the\ntotal number of combinatorial classes and the number of combinatorial classes\ncorresponding to loci of a specific (real) dimension q in parameter space, for\nfixed degree d. These results are extensions of a result by Douady, Estrada,\nand Sentenac, which shows that the number of combinatorial classes of the\nstructurally stable complex polynomial vector fields of degree d in the complex\nplane is the Catalan number C(d-1). We show that enumerating the combinatorial\nclasses is equivalent to a so-called bracketing problem. Then we analyze the\ngenerating functions and find closed-form expressions for the number of\nclasses, as functions of d and q, and we furthermore make an asymptotic\nanalysis of these sequences for d tending to infinity.\n These results are also applicable to special classes of Abelian\ndifferentials, quadratic differentials with double poles, and singular\nholomorphic foliations of the plane.\n