An explicit linear estimate for the number of zeros of Abelian integrals

Gal Binyamini, Gal Dor

Research output: Contribution to journalArticlepeer-review

Abstract

An Abelian integral is the integral over the level curves of a Hamiltonian H of an algebraic form ω. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the degrees H and ω. Petrov and Khovanskii have shown that this number grows at most linearly with the degree of ω, but gave a purely existential bound. Binyamini, Novikov and Yakovenko have given an explicit bound growing doubly exponentially with the degree. We combine the techniques used in the proofs of these two results, to obtain an explicit bound on the number of zeros of Abelian integrals growing linearly with deg ω.

Original languageEnglish
Pages (from-to)1931-1946
Number of pages16
JournalNonlinearity
Volume25
Issue number6
DOIs
StatePublished - Jun 2012

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • General Physics and Astronomy
  • Mathematical Physics

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