Hurwitz numbers for real polynomials

  • Ilia Itenberg

    Sorbonne Université, Paris, and École Normale Supérieure, Paris, France
  • Dimitri Zvonkine

    Université de Versailles, France


We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points.We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.

Cite this article

Ilia Itenberg, Dimitri Zvonkine, Hurwitz numbers for real polynomials. Comment. Math. Helv. 93 (2018), no. 3, pp. 441–474

DOI 10.4171/CMH/440