Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations
Annales de l'Institut Fourier, Volume 45 (1995) no. 1, pp. 183-221.

We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.

Let g be an element of a Hardy field which has an asymptotic series expansion in x, e x and λ, where λ tends to zero at least as rapidly as some negative power of exp(e x ). If λ actually occurs in the expansion, then g cannot satisfy a first-order algebraic differential equation over (x).

Nous nous intéressons aux croissances asymptotiques des solutions des équations différentielles algébriques qui sont des éléments d’un corps de Hardy, c’est-à-dire des solutions qui n’ont aucune composante oscillatoire. Nous prouvons qu’un développement imbriqué d’une telle solution ne peut tendre plus vite vers zéro qu’une vitesse fixe déterminée par l’ordre de l’équation différentielle. Nous considérons aussi les développements en série asymptotique généralisée, et obtenons, par exemple, le résultat suivant.

Soit g un élément d’un corps de Hardy qui a un développement en série avec fonctions de base x, e x et λ, où λ tend vers zéro au moins aussi vite qu’une puissance négative de exp(e x ). Si λ apparaît vraiment dans le développement, il s’ensuit que g ne peut satisfaire une équation différentielle du premier ordre sur (x).

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     title = {Growth orders occurring in expansions of {Hardy-field} solutions of algebraic differential equations},
     journal = {Annales de l'Institut Fourier},
     pages = {183--221},
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     year = {1995},
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Shackell, John. Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations. Annales de l'Institut Fourier, Volume 45 (1995) no. 1, pp. 183-221. doi : 10.5802/aif.1453. https://aif.centre-mersenne.org/articles/10.5802/aif.1453/

[1] N. Bourbaki, Éléments de Mathématiques, Ch. V : Fonctions d'une variable réelle. Appendice, pp. 36-55, Hermann, Paris, Second edition, 1961.

[2] N.G. De Bruijn, Asymptotic Methods in Analysis, North Holland, Amsterdam, 1958. | MR: 20 #6003 | Zbl: 0082.04202

[3] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice Hall Inc., Englewood Cliffs, NJ, 1965. | MR: 31 #4927 | Zbl: 0141.08601

[4] A. Lightstone and A. Robinson, Nonarchimedean Fields and Asymptotic Expansions, Elsevier, New York, 1975. | MR: 54 #2457 | Zbl: 0303.26013

[5] F. Olver, Asymptotics and Special Functions, Academic Press, 1974. | MR: 55 #8655 | Zbl: 0303.41035

[6] A. Robinson, On the real closure of a Hardy field, in “Theory of Sets and Topology”, G. Asser et al. eds., Deut. Verlag Wissenschaften, Berlin, 1972. | MR: 49 #4980 | Zbl: 0298.02061

[7] M. Rosenlicht, Hardy fields, J. Math. Anal. App., 93 (1983), 297-311. | MR: 85d:12001 | Zbl: 0518.12014

[8] M. Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc., 280 (1983), 659-671. | MR: 85d:12002 | Zbl: 0536.12015

[9] M. Rosenlicht, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc., 284 (1984), 829-836. | MR: 85i:12008 | Zbl: 0544.34052

[10] M. Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc., 299 (1987), 261-272. | MR: 88b:12010 | Zbl: 0619.34057

[11] L. A. Rubel, A universal differential equation, Bull. Amer. Math. Soc., 4 (1981), 345-349. | MR: 82e:34015 | Zbl: 0471.34008

[12] J.R. Shackell, Growth estimates for exp-log functions, J. Symbolic Comp., 10 (1990), 611-632. | MR: 92c:26001 | Zbl: 0727.26002

[13] J.R. Shackell, Limits of Liouvillian functions, Technical report, University of Kent at Canterbury, England, 1991.

[14] J.R. Shackell, Rosenlicht fields, Trans. Amer. Math. Soc., 335 (1993), 579-595. | MR: 93d:12012 | Zbl: 0772.34044

[15] J.R. Shackell, Zero-equivalence in function fields defined by algebraic differential equations, Trans. Amer. Math. Soc., 336 (1993), 151-171. | MR: 93e:12004 | Zbl: 0772.13012

[16] J.R. Shackell and B. Salvy, Asymptotic forms and algebraic differential equations, Technical report, University of Kent at Canterbury, England, 1993.

[17] W. Strodt, A differential algebraic study of the intrusion of logarithms into asymptotic expansions, in Contributions to Algebra, H. Bass, P.J. Cassidy and J. Kovacic eds., Academic Press, 1977, 355-375. | MR: 58 #623 | Zbl: 0372.12024

[18] W. Strodt and R.K. Wright, Asymptotic behaviour of solutions and adjunction fields for nonlinear first order differential equations, Mem. Amer. Math. Soc., 109 (1971). | MR: 44 #1884 | Zbl: 0235.34005

[19] B.J. Van Der Waerden, Algebra, vol. 2, Frederik Ungar Pub. Co., 1950. | MR: 12,236f | Zbl: 0137.25403

[20] H. Whitney, Complex Analytic Varieties, Addison-Wesley, 1972. | MR: 52 #8473 | Zbl: 0265.32008

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