Hurwitz numbers for real polynomials

Itenberg, Ilia; Zvonkine, Dimitri

doi:10.4171/cmh/440

Cited by 9 publications

(14 citation statements)

References 13 publications

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“…Hurwitz numbers are a widely studied subject, seen as central to combinatorial algebraic geometry. For basics see [4,5,15,20,23] and the references therein.…”

Section: Hurwitz Combinatoricsmentioning

confidence: 99%

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Recovery of Plane Curves from Branch Points

Agostini¹,

Markwig²,

Nollau³

et al. 2022

Preprint

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We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond.

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“…Hurwitz numbers are a widely studied subject, seen as central to combinatorial algebraic geometry. For basics see [4,5,15,20,23] and the references therein.…”

Section: Hurwitz Combinatoricsmentioning

confidence: 99%

“…We now turn to branched covers that are real. In the abstract setting of Hurwitz numbers H d , this has been studied in [3,15,20]. A cover f : C → P 1 is called real if the Riemann surface C has an involution which is compatible with complex conjugation on the Riemann sphere P 1 .…”

Section: Type Real?mentioning

confidence: 99%

Recovery of Plane Curves from Branch Points

Agostini¹,

Markwig²,

Nollau³

et al. 2022

Preprint

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“…This method is also valid in the study of counting covers. Itenberg and Zvonkine [13] found such a signed count of real polynomials Date: October 5, 2020. 2010 Mathematics Subject Classification.…”

Section: Introductionmentioning

confidence: 97%

On the lower bounds for real double Hurwitz numbers

Ding¹

2020

Preprint

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The real double Hurwitz number counts ramified covers of Riemann sphere with compatible involutions satisfying certain ramification data. J. Rau established a lower bound for these numbers in [19] using tropical covers with odd multiplicity. We improve Rau's lower bound by adding some tropical covers with even multiplicity. As an application, we prove the logarithmic equivalence of real and classical Hurwitz numbers without the existence assumption of Rau's lower bound.

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“…This is consistent with the best known results for Welschinger invariants and the Hurwitz-type counts of polynomials and rational functions mentioned before. For better comparison, let us recall the main asymptotic statements from [IZ16;ER17]. Let S pol (λ 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…x−p , f ∈ R[x], p ∈ R, respectively, with prescribed critical levels and ramification profiles as defined in [IZ16;ER17]. Set s pol (m) = S pol ((λ 1 , 1 2m ), .…”

Section: Introductionmentioning

confidence: 99%

Lower bounds and asymptotics of real double Hurwitz numbers

Rau

2019

Math. Ann.

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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).

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Hurwitz numbers for real polynomials

Cited by 9 publications

References 13 publications

Recovery of Plane Curves from Branch Points

Recovery of Plane Curves from Branch Points

On the lower bounds for real double Hurwitz numbers

Lower bounds and asymptotics of real double Hurwitz numbers

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