Signed counts of real simple rational functions
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Signed counts of real simple rational functions Boulos El Hilany1
· Johannes Rau2
Received: 8 January 2018 / Accepted: 24 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We study the problem of counting real simple rational functions ϕ with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus 0). We introduce a signed count of such functions which is independent of the position of the branch points, thus providing a lower bound for the actual count (which does depend on the position). We prove (non-)vanishing theorems for these signed counts and study their asymptotic growth when adding further simple branch points. The approach is based on Itenberg and Zvonkine (Comment Math Helv 93(2), 441–474, 2018) which treats the polynomial case. Keywords Enumerative problems · Real Hurwitz numbers · Dessins d’enfants · Signed counts Mathematics Subject Classification Primary 14N10 · 14H57; Secondary 05A15 · 14H30 · 14P99
1 Introduction A simple rational function of degree d is a function ϕ : CP1 → CP1 which, in affine coordinates, has the form ϕ(z) =
ψ(z) z−p
For this work, Boulos El Hilany and Johannes Rau were supported by the DFG Research Grant RA 2638/2-1.
B
Boulos El Hilany [email protected] Johannes Rau [email protected]
1
´ Instytut Matematyczny Polskiej Akademii Nauk, ul. Sniadeckich, 00-656 Warsaw, Poland
2
Universität Tübingen, Geschwister-Scholl-Platz, 72074 Tübingen, Germany
123
Journal of Algebraic Combinatorics
with ψ ∈ C[z], deg(ψ) = d, p ∈ C and ψ( p) = 0. We call ϕ real if ψ ∈ R[z] and p ∈ R, and increasing if the leading coefficient of ψ is positive. Two such functions are considered equivalent if they differ by linear coordinate change z → λz + μ, λ, μ ∈ R, λ > 0. A unique representative in each equivalence class of increasing functions is given by normalized functions, which are of the form ϕ(z) = ψ(z)/z, where ψ(0) = 0 and the leading coefficient of ψ is 1. Let b1 , . . . , bk ∈ C be the critical values (also called branch points) of ϕ and let Λ1 , . . . , Λk be the corresponding ramification profiles. This means Λ j = (Λ1j ≥ l
· · · ≥ Λ jj ) is the partition of d such that b j has l j preimages at which f locally takes the form z → z Λ j . We call {(b j , Λ j )} j=1,...,k the ramification data of ϕ. Let us now fix k distinct real points P = (b1 , . . . , bk ), b j ∈ R and k partitions of d = (Λ1 , . . . , Λk ) and set d = i, j (Λij − 1). We are interested in the number of normalized real simple rational functions f of degree d with ramification data {(b j , Λ j )} j=1,...,k . We denote the set of such functions by S(P, ). The number |S(P, )| does not depend on the position of the branch points, as long as we do not change their order of appearance on the real line. However, in general |S(P, )| is not invariant under a permutation of this order. To remedy the situation, we will define a sign ε(ϕ) ∈ {±1} for any ϕ ∈ S(P, ) (see Definition 1.5) such that the following
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