Polynomial bounds for the oscillation of solutions of Fuchsian systems
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2891-2926.

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.

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

Received:
Accepted:
DOI: 10.5802/aif.2511
Classification: 34M10,  34C08,  14Q20,  32S40
Keywords: Fuchsian systems, oscillation, zeros, semialgebraic varieties, effective algebraic geometry, monodromy
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Binyamini, Gal; Yakovenko, Sergei. Polynomial bounds for the oscillation  of solutions of Fuchsian systems. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2891-2926. doi : 10.5802/aif.2511. https://aif.centre-mersenne.org/articles/10.5802/aif.2511/

[1] Basu, Saugata; Vorobjov, Nicolai On the number of homotopy types of fibres of a definable map, J. Lond. Math. Soc. (2), Tome 76 (2007) no. 3, pp. 757-776 | Article | MR: 2377123 | Zbl: 1131.14060

[2] Binyamini, G.; Novikov, D.; Yakovenko, S. On the number of zeros of Abelian integrals. A constructive solution of the Infinitesimal Hilbert Sixteenth Problem, 2008 (Preprint, ArXiv:0808.2952 [math.DS], p. 1-48, to appear in Inventiones Mathematicae)

[3] Glutsyuk, A. A. Upper bounds of topology of complex polynomials in two variables, Mosc. Math. J., Tome 5 (2005) no. 4, p. 781-828, 972 | MR: 2266460 | Zbl: pre05182613

[4] Glutsyuk, A. A. An explicit formula for period determinant, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 4, pp. 887-917 | Article | Numdam | MR: 2266882 | Zbl: 1140.32011

[5] Glutsyuk, A. A.; Ilyashenko, Y. The restricted infinitesimal Hilbert 16th problem, Dokl. Akad. Nauk, Tome 407 (2006) no. 2, pp. 154-159 | MR: 2348308

[6] Glutsyuk, A. A.; Ilyashenko, Y. Restricted version of the infinitesimal Hilbert 16th problem, Mosc. Math. J., Tome 7 (2007) no. 2, p. 281-325, 351 | MR: 2337884 | Zbl: 1134.34019

[7] Grigor’ev, D. Y.; Vorobjov, N. N. Jr. Solving systems of polynomial inequalities in subexponential time, J. Symbolic Comput., Tome 5 (1988) no. 1-2, pp. 37-64 | Article | MR: 949112 | Zbl: 0662.12001

[8] Grigoriev, A. Singular perturbations and zeros of Abelian integrals (2001) (Ph. D. Thesis)

[9] Grigoriev, A. Uniform asymptotic bound on the number of zeros of Abelian integrals, 2003 (Preprint ArXiv:math.DS/0305248)

[10] Heintz, Joos; Roy, Marie-Françoise; Solernó, Pablo Sur la complexité du principe de Tarski-Seidenberg, Bull. Soc. Math. France, Tome 118 (1990) no. 1, pp. 101-126 | EuDML: 87591 | Numdam | MR: 1077090 | Zbl: 0767.03017

[11] Ilyashenko, Y. Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.), Tome 39 (2002) no. 3, p. 301-354 (electronic) | Article | MR: 1898209 | Zbl: 1004.34017

[12] Ilyashenko, Y. Some open problems in real and complex dynamical systems, Nonlinearity, Tome 21 (2008) no. 7, p. T101-T107 | Article | MR: 2425322 | Zbl: pre05305649

[13] Ilyashenko, Y.; Yakovenko, S. Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, Tome 86, American Mathematical Society, Providence, RI, 2008 | MR: 2363178 | Zbl: pre05223651

[14] Khovanskiĭ, A. G. Real analytic manifolds with the property of finiteness, and complex abelian integrals, Funktsional. Anal. i Prilozhen., Tome 18 (1984) no. 2, pp. 40-50 | Article | MR: 745698 | Zbl: 0584.32016

[15] Khovanskiĭ, A. G. Fewnomials, Translations of Mathematical Monographs, Tome 88, American Mathematical Society, Providence, RI, 1991 (Translated from the Russian by Smilka Zdravkovska) | MR: 1108621 | Zbl: 0728.12002

[16] Levin, B. Ja. Distribution of zeros of entire functions, Translations of Mathematical Monographs, Tome 5, American Mathematical Society, Providence, R.I., 1980 (Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman) | MR: 589888 | Zbl: 0152.06703

[17] Novikov, D. Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions, Proc. Amer. Math. Soc., Tome 129 (2001) no. 12, p. 3753-3755 (electronic) | Article | MR: 1860513 | Zbl: 0983.34019

[18] Novikov, D.; Yakovenko, S. Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems, Electron. Res. Announc. Amer. Math. Soc., Tome 5 (1999), p. 55-65 (electronic) | Article | EuDML: 223357 | MR: 1679454 | Zbl: 0922.58076

[19] Novikov, D.; Yakovenko, S. Redundant Picard-Fuchs system for abelian integrals, J. Differential Equations, Tome 177 (2001) no. 2, pp. 267-306 | Article | MR: 1876646 | Zbl: 1011.37042

[20] Roitman, M.; Yakovenko, S. On the number of zeros of analytic functions in a neighborhood of a Fuchsian singular point with real spectrum, Math. Res. Lett., Tome 3 (1996) no. 3, pp. 359-371 | MR: 1397684 | Zbl: 0871.34005

[21] Varchenko, A. N. Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles, Funktsional. Anal. i Prilozhen., Tome 18 (1984) no. 2, pp. 14-25 | Article | MR: 745696 | Zbl: 0545.58038

[22] Yakovenko, Sergei On functions and curves defined by ordinary differential equations, The Arnoldfest (Toronto, ON, 1997) (Fields Inst. Commun.) Tome 24, Amer. Math. Soc., Providence, RI, 1999, pp. 497-525 | MR: 1733590 | Zbl: 0949.34023

[23] Yakovenko, Sergei Quantitative theory of ordinary differential equations and the tangential Hilbert 16th problem, On finiteness in differential equations and Diophantine geometry (CRM Monogr. Ser.) Tome 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41-109 | MR: 2180125 | Zbl: 1104.34025

[24] Yakovenko, Sergei Oscillation of linear ordinary differential equations: on a theorem of A. Grigoriev, J. Dyn. Control Syst., Tome 12 (2006) no. 3, pp. 433-449 | Article | MR: 2233029 | Zbl: 1131.34028

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