Polynomial bounds for the oscillation of solutions of Fuchsian systems

Binyamini, Gal; Yakovenko, Sergei

Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2891-2926.

Nous étudions le problème d’une borne supérieure effective sur le nombre des racines isolées des solutions de systèmes de type Fuchs sur la sphère de Riemann. Le résultat principal est une borne explicite non uniforme à croissance polynômiale sur la frontière de l’ensemble des systèmes fuchsiens de dimension $n$ quelconque ayant m singularités. Comme une fonction de $n,m$, la borne est doublement exponentielle dans le sens précis décrit dans le manuscrit.

Comme corollaire, nous obtenons la solution à croissance polynômiale du problème d’Hilbert infinitésimal restreint, qui améliore les bornes exponentielles récemment obtenues par A. Glutsyuk et Yu. Ilyashenko

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.

Classification : 34M10, 34C08, 14Q20, 32S40
Keywords: Fuchsian systems, oscillation, zeros, semialgebraic varieties, effective algebraic geometry, monodromy
Mot clés : systèmes fuschiens, oscillation, zéro, variétés semi-algébriques, monodromie

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