Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group
type: Item
instance of: scholarly article
title: Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group
zbMATH Open document ID: 0865.34007
DOI: 10.1007/BF01884313
published in: Inventiones Mathematicae
publication date: 1996-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/144312
review text: The paper contains a detailed discussion of the general fact that if the functions \(y_1(t)\), \(y_2(t)\), \(y_n(t)\), real analytic on \([a,b]\), constitute a fundamental system of solutions for a linear differential equation with real meromorphic coefficients and an irreducible monodromy group, then the number of isolated zeroes on \([a,b]\), of any combination \(f(t)=\sum^n_{j,r=1} P_{jr}(t)Y^{(r-1)}_j(t)\) with real polynomial coefficients \(P_{jr}(t)\) of degree \(\leq d\), can be at most \(2^{2^{O(d)}}\) with constant \(O(d)\) depending on the linear equation and the segment \([a,b]\) only. Then, it is proved that for a generic real polynomial \(H(x,y)\) in two variables and a polynomial 1-form \(\omega=P(x,y)dx+Q(x,y)dy\) of degree \(d=\max(\deg P,\deg Q)\) the number of real zeroes of the complete Abelian integral \(I(t)=\phi\omega\) over ovals of the curve \(\{H=t\}\) on any compact segment \(t\in(a,b)\subset\mathbb{R}\) not containing critical points of the Hamiltonian \(H\) does not exceed \(2^{2^{O(d)}}\) as \(d\to\infty\) uniformly over all forms.
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 33E99
zbMATH DE Number: 871786
zbMATH Keywords: complete Abelian integrals
zbMATH Keywords: irreducible monodromy groups
zbMATH Keywords: exponential estimates
author: Yu Ilyashenko
author: S. Yakovenko
reviewed by: Mahadeb Dutta
MaRDI profile type: Publication
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OpenAlex ID: W2314046183
description: scientific article; zbMATH DE number 871786