A SIMPLE GEOMETRICAL CONDITION FOR THE EXISTENCE OF n-SQUARE ROOTS OF MEROMORPHIC MAPS

Abstract:

Let S be a connected Riemann surface and let ϕ: S→ ̂C be a surjective meromorphic map with finite branch set Bϕ and such that∞, 0∈ Bϕ. If n≥ 2, then we describe a simple geometrical necessary and sufficient condition for the existence of some n-square root, that is, a meromorphic map ψ: S→ ̂C such that ϕ= ψn. 1. introduction Let S be a connected (not necessarily compact) Riemann surface and let ϕ: S→ ̂C be a surjective meromorphic map onto the Riemann sphere ̂C with finite branch set Bϕ⊂ ̂C (we say that ϕ is of finite type) and∞, 0∈ Bϕ. We also assume that ϕ is of bounded local degree, that is, if mϕ (q)≥ 1 is the local degree of ϕ at q∈ S, then the set Mp:={mϕ (q): q∈ ϕ− 1 (p)} is bounded for every p∈ Bϕ. In this situation, we may define the branch order of p∈ Bϕ as the minimum common multiple of the values in Mp. By an n-square root of ϕ, where n≥ 2, we mean a meromorphic map ψ: S→ ̂C such that ϕ= ψn. The existence of such an n-square root implies necessarily that∞ and 0 both have branch order a multiple of n. In [2], it was provided a simple geometrical necessary and sufficient condition for ϕ (of finite type and bounded local degree) two have a 2-square root. The aim of this paper is to obtain a similar geometrical condition for the existence of an n-square root (Theorem 1). Before, we need some definitions. An admisible arc for ϕ is a simple arc δ⊂ ̂C whose end points are 0 and∞, and Bϕ\{0,∞}⊂ δ. In this case, the graph ϕ− 1 (δ)= Gδ⊂ S defines a map Fδ on S (that is, a 2-cell decomposition), where each of it faces is a polygon with 2 (r− 1) sides. We say that Fδ is nZ-orientable if we may label its faces using …

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