Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

John Shackell

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 1, page 183-221
  • ISSN: 0373-0956

Abstract

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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 ℝ ( x ) .

How to cite

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Shackell, John. "Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations." Annales de l'institut Fourier 45.1 (1995): 183-221. <http://eudml.org/doc/75113>.

@article{Shackell1995,
abstract = {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 $\lambda $, where $\lambda $ tends to zero at least as rapidly as some negative power of $\exp (e^x)$. If $\lambda $ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over $\{\Bbb R\}(x)$.},
author = {Shackell, John},
journal = {Annales de l'institut Fourier},
keywords = {asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion},
language = {eng},
number = {1},
pages = {183-221},
publisher = {Association des Annales de l'Institut Fourier},
title = {Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations},
url = {http://eudml.org/doc/75113},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Shackell, John
TI - Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 1
SP - 183
EP - 221
AB - 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 $\lambda $, where $\lambda $ tends to zero at least as rapidly as some negative power of $\exp (e^x)$. If $\lambda $ actually occurs in the expansion, then $g$ cannot satisfy a first-order algebraic differential equation over ${\Bbb R}(x)$.
LA - eng
KW - asymptotic growth of Hardy-field solutions; algebraic differential equations; asymptotic series expansion
UR - http://eudml.org/doc/75113
ER -

References

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  1. [1] N. BOURBAKI, Éléments de Mathématiques, Ch. V : Fonctions d'une variable réelle. Appendice, pp. 36-55, Hermann, Paris, Second edition, 1961. 
  2. [2] N.G. de BRUIJN, Asymptotic Methods in Analysis, North Holland, Amsterdam, 1958. Zbl0082.04202MR20 #6003
  3. [3] R.C. GUNNING and H. ROSSI, Analytic Functions of Several Complex Variables, Prentice Hall Inc., Englewood Cliffs, NJ, 1965. Zbl0141.08601MR31 #4927
  4. [4] A. LIGHTSTONE and A. ROBINSON, Nonarchimedean Fields and Asymptotic Expansions, Elsevier, New York, 1975. Zbl0303.26013MR54 #2457
  5. [5] F. OLVER, Asymptotics and Special Functions, Academic Press, 1974. Zbl0303.41035MR55 #8655
  6. [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. Zbl0298.02061MR49 #4980
  7. [7] M. ROSENLICHT, Hardy fields, J. Math. Anal. App., 93 (1983), 297-311. Zbl0518.12014MR85d:12001
  8. [8] M. ROSENLICHT, The rank of a Hardy field, Trans. Amer. Math. Soc., 280 (1983), 659-671. Zbl0536.12015MR85d:12002
  9. [9] M. ROSENLICHT, Rank change on adjoining real powers to Hardy fields, Trans. Amer. Math. Soc., 284 (1984), 829-836. Zbl0544.34052MR85i:12008
  10. [10] M. ROSENLICHT, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc., 299 (1987), 261-272. Zbl0619.34057MR88b:12010
  11. [11] L. A. RUBEL, A universal differential equation, Bull. Amer. Math. Soc., 4 (1981), 345-349. Zbl0471.34008MR82e:34015
  12. [12] J.R. SHACKELL, Growth estimates for exp-log functions, J. Symbolic Comp., 10 (1990), 611-632. Zbl0727.26002MR92c:26001
  13. [13] J.R. SHACKELL, Limits of Liouvillian functions, Technical report, University of Kent at Canterbury, England, 1991. Zbl0845.26001
  14. [14] J.R. SHACKELL, Rosenlicht fields, Trans. Amer. Math. Soc., 335 (1993), 579-595. Zbl0772.34044MR93d:12012
  15. [15] J.R. SHACKELL, Zero-equivalence in function fields defined by algebraic differential equations, Trans. Amer. Math. Soc., 336 (1993), 151-171. Zbl0772.13012MR93e:12004
  16. [16] J.R. SHACKELL and B. SALVY, Asymptotic forms and algebraic differential equations, Technical report, University of Kent at Canterbury, England, 1993. Zbl0843.68045
  17. [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. Zbl0372.12024MR58 #623
  18. [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). Zbl0235.34005MR44 #1884
  19. [19] B.J. van der WAERDEN, Algebra, vol. 2, Frederik Ungar Pub. Co., 1950. Zbl0137.25403MR12,236f
  20. [20] H. WHITNEY, Complex Analytic Varieties, Addison-Wesley, 1972. Zbl0265.32008MR52 #8473

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