# zbMATH — the first resource for mathematics

Algebraic families of analytic functions. I. (English) Zbl 0886.34005
There are investigated formal power series $y(x)= \sum_{n=1}^\infty a_k(\lambda) x^n$ as algebraic functions satisfying $$y=p_0(x)+ p_1(x)y+\dots+ p_d(x)y^d$$ or in an analogous way as solutions of polynomial differential equations $y'= p_0(x)+ p_1(x)y+\dots+ p_d(x)y^d,$ $$\lambda= (\lambda_{ij})\in \mathbb{C}^{(d+1)}$$ is a tupel of parameters.
By recurrence formulae one gets certain estimations for the polynomial coefficients $$a_k(\lambda)$$.
This leads to the membership of the regarded algebraic functions to so-called Bernstein classes and to statements on the numbers of zeros.
Furthermore these constructions are in an algebraic context related to so-called $$A_0$$-series consisting of all series $$f_\lambda(x)= \sum_{n=0}^\infty a_n(x)x^n$$, where the degree of $$a_j(\lambda)$$ increases linearly in $$j$$ and the norm $$|a_j(\lambda)|$$ increases exponentially. The ideal generated from all $$a_k(\lambda)$$ is the so-called Bautin ideal of $$f_\lambda$$, and properties of the above series $$y(x)$$ are related to this ideal. The last part of the paper concerns Bautin ideals of solutions of higher order linear differential equations.

##### MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
Full Text:
##### References:
 [1] Bautin, N.N., 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, 1-19, (1954) [2] Amer. math. soc. trans. series I, 5, 396-413, (1962) [3] E. Bierstone, P. Milman, 1988, The local geometry of analytic mappings, ETS Editrice, Pisa [4] M. Briskin, Y. Yomdin, Algebraic families of analytic functions, II · Zbl 0886.34005 [5] Cherkas, L.A., Number of limit cycles of an autonomous second-order system, J. differential equations, 5, 666-668, (1976) · Zbl 0365.34039 [6] Chavarriga, J., Integrable systems in the plane with a center type linear part, Appl. math., 22, 285-309, (1994) · Zbl 0809.34002 [7] Chicone, C.; Jacobs, M., Bifurcations of critical periods for plane vector fields, Trans. amer. math. soc., 312, 433-486, (1989) · Zbl 0678.58027 [8] Fefferman, C.; Narasimhan, R., Bernstein’s inequality on algebraic curves, Ann. inst. Fourier, Grenoble, 43, 1319-1348, (1993) · Zbl 0842.26013 [9] Fefferman, C.; Narasimhan, R., On the polynomial-like behaviour of certain algebraic functions, Ann. inst. Fourier, Grenoble, 44, 1091-1179, (1994) · Zbl 0811.14046 [10] Francoise, J.-P.; Pugh, C.C., Keeping track of limit cycles, J. differential equations, 65, 139-157, (1986) · Zbl 0602.34019 [11] J.-P. Francoise, Y. Yomdin, Bernstein inequality and applications to analytic geometry and differential equations, J. Funct. Anal. · Zbl 0869.34008 [12] Gabrielov, A., Formal relations between analytic functions, USSR izv., 7, 1056-1088, (1973) · Zbl 0297.32007 [13] A. Gasull, A. Guillamon, V. Mañosa, 1995, Centre and isochronicity conditions for systems with homogeneous nonlinearities · Zbl 0909.34030 [14] Gavrilov, L.; Hayman, W.K., Isochronism of plane polynomial Hamiltonian systems, Pacific J. math., 44, 117-137, (1995) [15] Il’yashenko, Yu., Divergence of the linearizing series, Funktsional anal. i prilozhen, 13, 87-88, (1979) [16] Il’yashenko, Yu.; Yakovenko, S., Counting real zeroes of analytic functions, satisfying linear ordinary differential equations, J. differential equations, 126, 87-105, (1996) · Zbl 0847.34010 [17] Il’yashenko, Yu.; Yakovenko, S., Double exponential estimate for the number of real zeroes of complete abelian integrals, Inventiones mathematicae, 121, 613-650, (1995) · Zbl 0865.34007 [18] Laine, I., Nevanlinna theory and complex differential equations, De gruyter studies in math., 15, (1993), Walter de Gruyter Berlin/New York [19] Lipshitz, L.; Rubel, L.A., A gap theorem for power series solutions of algebraic differential equations, Amer. J. math., 108, 1193-1214, (1986) · Zbl 0605.12014 [20] Mahler, K., Lectures on transcendental numbers, Lnm 546, (1976), Springer-Verlag Berlin/Heidelberg/New York [21] Novikov, D.; Yakovenko, S., Simple exponential estimate for the number of zeroes of complete abelian integrals, Ann. inst. Fourier, Grenoble, 45, 897-927, (1995) · Zbl 0832.58028 [22] Roussarie, R., A note on finite cyclicity and Hilbert’s 16th problem, Lnm 1331, (1988), Springer-Verlag New York/Berlin, p. 161-168 · Zbl 0676.58046 [23] N. Roytvarf, Y. Yomdin, Bernstein’s classes, Ann. Inst. Fourier, Grenoble [24] Siegel, C.L., Transcendental numbers, (1949), Princeton Univ. Press Princeton · Zbl 0039.04402 [25] Yomdin, Y., Local complexity growth for iterations of real analytic mappings and semi-continuity moduli of the entropy, Erg. th. and dynam. syst., 11, 583-602, (1991) · Zbl 0756.58041 [26] Y. Yomdin, Oscillation of analytic curves, Proc. Amer. Math. Soc. · Zbl 0897.32001 [27] Žoladek, H., On certain generalization of Bautin’s theorem, Nonlinearity, 7, 273-280, (1994) · Zbl 0838.34035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.