Polynomial bounds for the oscillation of solutions of Fuchsian systems

Gal Binyamini[1]; Sergei Yakovenko[1]

  • [1] Weizmann Institute of Science Rehovot 76100 (Israël)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2891-2926
  • ISSN: 0373-0956

Abstract

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We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension n having m singular points. As a function of n , m , this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeroes of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.

How to cite

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Binyamini, Gal, and Yakovenko, Sergei. "Polynomial bounds for the oscillation of solutions of Fuchsian systems." Annales de l’institut Fourier 59.7 (2009): 2891-2926. <http://eudml.org/doc/10475>.

@article{Binyamini2009,
abstract = {We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension $n$ having $m$ singular points. As a function of $n,m$, this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeroes of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.},
affiliation = {Weizmann Institute of Science Rehovot 76100 (Israël); Weizmann Institute of Science Rehovot 76100 (Israël)},
author = {Binyamini, Gal, Yakovenko, Sergei},
journal = {Annales de l’institut Fourier},
keywords = {Fuchsian systems; oscillation; zeros; semialgebraic varieties; effective algebraic geometry; monodromy},
language = {eng},
number = {7},
pages = {2891-2926},
publisher = {Association des Annales de l’institut Fourier},
title = {Polynomial bounds for the oscillation of solutions of Fuchsian systems},
url = {http://eudml.org/doc/10475},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Binyamini, Gal
AU - Yakovenko, Sergei
TI - Polynomial bounds for the oscillation of solutions of Fuchsian systems
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2891
EP - 2926
AB - We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension $n$ having $m$ singular points. As a function of $n,m$, this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeroes of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.
LA - eng
KW - Fuchsian systems; oscillation; zeros; semialgebraic varieties; effective algebraic geometry; monodromy
UR - http://eudml.org/doc/10475
ER -

References

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