Lower bounds and asymptotics of real double Hurwitz numbers

Abstract

We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on the distribution of branch points. We use it to prove, under certain conditions, existence of real Hurwitz covers as well as logarithmic equivalence of real and classical Hurwitz numbers. The lower bound is based on the "tropical" computation of real Hurwitz numbers in [MR15]. branch points lie in RP 1 , and 0 ≤ p ≤ r denotes the number of simple branch points on the positive half axis of RP 1 \ {0, ∞}.In this paper, we define numbers Z g (λ , µ) such thatThe main results of this paper state that, under certain conditions, these lower bounds are non-zero and have logarithmic asymptotic growth equal to H C g (λ , µ) (see Proposition 5.2, Theorem 5.7, Theorem 5.10). The definition of Z g (λ , µ) is based on the tropical computation of real double Hurwitz numbers in [MR15].Note that the real double Hurwitz numbers H R g (λ , µ; p) indeed depend on p, or in other words, on the position of the branch points. This is the typical behaviour of enumerative problems over R (instead of C). It is therefore of interest to find lower bounds for real enumerative problems (with respect to the choice of conditions, the branch points here) and use these bounds to prove existence of real solutions or to compare the number of real and complex solutions of the problem. Such investigations have been carried out e.g. for real Schubert calculus [Sot97; MT16], counts of algebraic curves in surfaces passing through points [Wel05; IKS04] (see also [GZ15]) and counts of polynomials/simple rational functions with given critical levels [IZ16; ER17]. In most of these examples, a lower bound is constructed by defining a signed count of the real solutions (i.e., each real solution is counted with +1 or −1 according to some rule) and showing that this signed count is invariant under change of the conditions. In this paper, we prove similar results for double Hurwitz numbers without the explicit constructions of a signed count. We hope that this rather simple approach can be extended to other situations using sufficiently nice combinatorial descriptions of the counting problem.One way of defining Z g (λ , µ) is as follows: It is the number of those tropical covers which contribute to the tropical count of H R g (λ , µ; p) with odd multiplicity. We prove in Theorem 4.10 that these numbers provide lower bounds for H R g (λ , µ; p) as explained above. Next, we give exact numerical criteria on λ , µ in order for Z g (λ , µ) to be non-zero, proving existence of real Hurwitz covers in these case (see Proposition 5.1, Proposition 5.2 and Remark 5.4).

On the lower bounds for real double Hurwitz numbers

“…Remark 2.13. We use the real multiplicity introduced in [19] which differs from the one in [17]. The real multiplicity of a tropical cover in [17] allows T to contain symmmetric even forks.…”

“…The real multiplicity of a tropical cover in [17] allows T to contain symmmetric even forks. The readers may refer to [19,Remark 3.5] for more detail analysis on the difference between these two definitions.…”

“…The lower bound for H R g (λ, µ; s), 0 ≤ s ≤ r, established in [19] is the number of real tropical covers with odd multiplicity. These covers contribute to the tropical count of H R g (λ, µ; s), 0 ≤ s ≤ r, for any splitting of branch points x = x + ⊔ x − .…”

“…Based on the tropical interpretation of real double Hurwitz numbers in [17], Rau [19] found a lower bound for real double Hurwitz numbers by considering the tropical covers with odd multiplicity. Tropical covers with odd multiplicity are called zigzag covers (see Definition 3.1).…”

“…Remark 1.2. The assumption on odd elements in λ and µ in [19,Theorem 5.10] is a necessary condition to guarantee the existence of zigzag covers. When zigzag covers exist, Rau's result is consistent with the best known results for Welschinger invariants and Hurwitz type counts of polynomials and simple rational polynomials.…”