Bernstein classes

Roytwarf, N.; Yomdin, Yosef

Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 825-858.

Une des inégalités classiques de Bernstein compare les maximas d’un polynôme de degré donné sur l’intervalle [-1,1] et sur l’ellipse du plan complexe de foyers -1, 1 et de semi-axes $R$. Nous démontrons une inégalité similaire pour une branche de fonction algébrique de degré donné sur son disque maximal de régularité avec une constante donnée explicitement, dépendant seulement du degré. En particulier cela améliore une inégalité récente due à Fefferman et Narasimhan et cela répond à une de leurs questions. Nous présentons en détail diverses propriétés de classes de fonctions satisfaisant des inégalités du type de celle de Bernstein et diverses approches pour établir ces inégalités.

One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1, 1 and the semiaxes $R$. We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions. We present in detail various properties of the classes of functions, satisfying Bernstein type inequalities and various approaches to establishing such inequalities.

[1] V.I. Arnold, Yu. Il'Yashenko, Ordinary differential equations, Encyclopedia of Mathematical Sciences 1 (Dynamical Systems - I), Springer, Berlin, 1988. | Zbl

[2] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium state of the type focus or center, Amer. Math. Soc. Trans. 100 (1954) 1-19, Providence, R.I.; reprinted in: Stability and Dynamical Systems, Amer. Math. Soc. Trans. Series, I 5 (1962), 396-413. | Zbl

[3] S. Bernstein, Sur une propriété des polynômes, Proc. Kharkov Math. Society, Serie 2, v. 14 (1913), 1-6.

[4] M. Biernacki, Sur les fonctions multivalentes d'ordre p, C.R. Acad. Sci. (Paris), 203 (1936), 449-451. | JFM | Zbl

[5] L. Bos, N. Levenberg, P. Milman, B.A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of ℝN, Indiana Univ. Math. J., 44 (1995), 115-137. | MR | Zbl

[6] L. Bos, P. Milman, Sobolev-Gagliardo-Nirenberg and Markov Type Inequalities on Subanalytic Domains, Geometric and Functional Analysis, 5 (6) (1995), 853-923. | MR | Zbl

[7] M. Briskin, Y. Yomdin, Algebraic families of analytic functions, I, to appear, J. of Diff. Equations. | Zbl

[8] M. Briskin, Y. Yomdin, Algebraic families of analytic functions, II, in preparation. | Zbl

[9] Yu. Brudnyi, M. Ganzburg, On an extremal problem for polynomials of n variables, Math. USSR Izv., 37 (1973), 344-355. | Zbl

[10] A. Brudnyi, Bernstein-type inequality for algebraic functions, preprint, 1996. | Zbl

[11] L.A. Cherkas, Number of limit cycles of an autonomous second-order system, Differ. Eq., (1976), 666-668. | Zbl

[12] J. Chavarriga, Integrable systems in the plane with a center type linear part, Applicationes Mathematicae, 22 (1994), 285-309. | EuDML | MR | Zbl

[13] C. Chicone, M. Jacobs, Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312 (1989), 433-486. | MR | Zbl

[14] D.V. Chudnovsky, G.V. Chudnovsky, On expansions of algebraic functions in power and Puiseux series, I, J. of Complexity, 2 (1986), 271-294. | MR | Zbl

[15] D.V. Chudnovsky, G.V. Chudnovsky, On expansions of algebraic functions in power and Puiseux series, II, J. of Complexity, 3 (1987), 1-25. | MR | Zbl

[16] C. Fefferman, R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier, Grenoble, 43-5 (1993), 1319-1348. | EuDML | Numdam | MR | Zbl

[17] C. Fefferman, R. Narasimhan, On the polynomial-like behaviour of certain algebraic functions, Ann. Inst. Fourier, Grenoble, 44-2 (1994), 1091-1179. | EuDML | Numdam | MR | Zbl

[18] C. Fefferman, R. Narasimhan, A local Bernstein inequality on real algebraic varieties, preprint, 1995. | Zbl

[19] C. Fefferman, R. Narasimhan, Bernstein's inequality and the resolution of spaces of analytic functions, to appear in Duke Math. J. | Zbl

[20] J.-P. Francoise and C.C. Pugh, Keeping track of limit cycles, J. Diff. Equations, 65 (1986), 139-157. | MR | Zbl

[21] J.-P. Francoise and Y. Yomdin, Bernstein inequality and applications to analytic geometry and differential equations, to appear, J. of Functional Anal. | Zbl

[22] J.-P. Francoise, Y. Yomdin, Projection of analytic sets and Bernstein inequalities, preprint, 1996. | Zbl

[23] A. Gabrielov, Projections of semi-analytic sets, Funct. Anal. Appl., 2(4) (1968), 282-291. | MR | Zbl

[24] A. Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Res. Lett., 2 (1995), 1-15. | Zbl

[25] A. Gabrielov, Multiplicities of Pfaffian intersections and the Lojasiewicz inequality, Selecta Matematica, New Series, 11 (1995), 113-127. | MR | Zbl

[26] A. Gabrielov, Formal relations between analytic functions, USSR Izv., 7 (1973), 1056-1088. | Zbl

[27] A. Gasull, A. Guillamon, V. Mañosa, Centre and isochronicity conditions for systems with homogeneous nonlinearities, preprint, 1995. | MR | Zbl

[28] L. Gavrilov, Isochronism of plane polynomial Hamiltonian systems, Prepublication no. 49, Laboratoire de Topologie et Geometrie, Toulouse, 1995. | Zbl

[29] W.K. Hayman, Differential inequalities and local valency, Pacific J. of Math., 44(1) (1973), 117-137. | MR | Zbl

[30] Yu. Il'Yashenko, Divergence of the linearizing series, Funct. Anal. Appl., 13 (3) (1979), 87-88. | Zbl

[31] Yu. Il'Yashenko, S. Yakovenko, Counting real zeroes of analytic functions, satisfying linear ordinary differential equations, J. Diff. Equations, 126 (1) (1996), 87-105. | Zbl

[32] Yu. Il'Yashenko, S. Yakovenko, Double exponential estimate for the number of real zeroes of complete abelian integrals, Inventiones Mathematicae, 121 (1995), 613-650. | EuDML | Zbl

[33] A.G. Khovanski, Fewnomials, AMS Publ., Providence, RI, 1991. | MR

[34] I. Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Math., 15, Walter de Gruyter, Berlin, New York, 1993. | MR | Zbl

[35] K. Mahler, Lectures on transcendental numbers, LNM 546, Springer-Verlag, Berlin-Heidelberg-New York, 1976. | MR | Zbl

[36] A.L. Neto, On the number of solutions of the equation x´ = P(x,t), for which x(0) = x(1), Inventiones Math., 59 (1980), 67-76. | EuDML | MR | Zbl

[37] D. Novikov, S. Yakovenko, Simple exponential estimate for the number of zeroes of complete Abelian integrals, Ann. Inst. Fourier, Grenoble, 45-4 (1995), 897-927. | EuDML | Numdam | MR | Zbl

[38] G. Petrov, A. Khovanski, On a linear bound for the number of zeroes of abelian integrals, to appear.

[39] R. Roussarie, A note on finite cyclicity and Hilbert's 16th problem, LNM 1331, Springer, New York-Berlin (1988), 161-168. | MR | Zbl

[40] R. Roussarie, Cyclicité finie des lacets et des points cuspidaux, Nonlinearity, 2 (1989), 73-117. | MR | Zbl

[41] N. Roytvarf, Bernstein inequality for algebraic functions and for solutions of linear differential equations, Ph. D. thesis, Rehovot, 1996.

[42] N. Roytvarf, A Markov-type inequality for Wronskians, in preparation.

[43] N. Roytvarf, Counting zeroes of linear combinations of analytic functions, in preparation.

[44] N. Roytvarf, Taylor coefficients of solutions of linear differential equations with polynomial coefficients, in preparation.

[45] N. Roytvarf, Y. Yomdin, Bernstein's inequality for algebraic functions, in preparation.

[46] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR Izv., 20 (1983), 493-502. | Zbl

[47] C.L. Siegel, Transcendental numbers, Princeton University Press, Princeton, 1949. | MR | Zbl

[48] A.J. Van Der Poorten, On the number of zeroes of functions, Indag. Math. | Zbl

[49] M. Voorhoeve, A.J. Van Der Poorten, R. Tijdeman, On the number of zeroes of certain functions, Indag. Math., (1975), 407-416. | MR | Zbl

[50] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57, 3 (1987), 285-300. | MR | Zbl

[51] Y. Yomdin, Ck resolution of semialgebraic mappings, Israel J. Math., 57, 3 (1987), 301-317. | MR | Zbl

[52] Y. Yomdin, Local complexity growth for iterations of real analytic mappings and semi-continuity moduli of the entropy, Erg. Th. and Dynam. Syst., 11 (1991), 583-602. | MR | Zbl

[53] Y. Yomdin, Zeroes of analytic functions and intersection growth, in preparation.

[54] Y. Yomdin, Oscillation of analytic curves, preprint, 1995. | MR | Zbl

[55] Y. Yomdin, Global Bernstein inequality on algebraic curves, in preparation.

[56] H. Źoladek, On certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-280. | MR | Zbl