Download PDF - On the rings of formal solutions of polynomial differential equationsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On the rings of formal solutions of polynomial differential equations

Authors 1

Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Ciudad Universitaria de Canto Blanco, 28 049 Madrid, Spain

Abstract

The paper establishes the basic algebraic theory for the Gevrey rings. We prove the Hensel lemma, the Artin approximation theorem and the Weierstrass-Hironaka division theorem for them. We introduce a family of norms and we look at them as a family of analytic functions defined on some semialgebraic sets. This allows us to study the analytic and algebraic properties of this rings.

Bibliography

  1. [ADM] G. R. Allan, H. G. Dale, J. P. McClure Pseudo-Banach algebras Studia Math. 40 (1971), 55-69
  2. [AHV] J. M. Aroca, H. Hironaka, J. L. Vicente The Theory of the Maximal Contact Memorias de Matemática del Instituto 'Jorge Juan' 29, Madrid, 1975
  3. [BR] R. Benedetti, J. J. Risler Real Algebraic and Semi-algebraic Sets Actualités Math., Hermann, Paris, 1990
  4. [B] J. W. Brewer Power Series over Commutative Rings Lecture Notes in Pure and Appl. Math. 64, Marcel Dekker, New York, 1981
  5. [Ca] J. Cano On the series defined by differential equations with an extension of the Puiseux polygon construction to this series Analysis 13 (1993), 103-119
  6. [CC1] J. Chaumat, A. M. Chollet Sur le théorème de division de Weierstrass Studia Math. 116 (1995), 59-84
  7. [CC2] J. Chaumat, A. M. Chollet Théorème de preparation dans les classes ultradifferentiables C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1305-1310
  8. [Co] P. M. Cohn Puiseux's theorem revisited J. Pure Appl. Algebra 31 (1984), 1-4
  9. [G] M. Gevrey Sur la nature analytique des solutions des équations aux dérivées partielles Ann. Sci. École Norm. Sup. (3) 25 (1918), 129-190
  10. [H] H. Hironaka Idealistic exponents of singularity in: Algebraic Geometry, John Hopkins Univ. Press, Baltimore, 1977, 52-125
  11. [Mi] E. Maillet Sur les séries divergentes et les équations differentielles Ann. Sci. École Norm. Sup. 3 (1903), 487-518
  12. [Ml] B. Malgrange Sur le théorème de Maillet Asymptot. Anal. 2 (1989), 1-4
  13. [M] H. Matsumura Commutative Algebra Math. Lecture Note Ser. 56, Benjamin/Cumming Publishing Co., Reading, 1980
  14. [N] M. Nagata Local Rings Robert E. Krieger Publishing Co., Huntington, 1975
  15. [O] S. Ouchi Formal solutions with Gevrey type estimates of nonlinear partial differential equations J. Math. Sci. Univ. Tokyo 1 (1994), 205-237
  16. [R] C. Rotthaus On the approximation theory of excellent rings Invent. Math. 88 (1987), 39-63
  17. [T] J. Cl. Tougeron Sur les ensembles semi-analytiques avec conditions Gevrey au bord Ann. Sci. École Norm. Sup. (4) 27 (1994), 173-208
  18. [Z1] M. A. Zurro Le théorème de division pour les séries Gevrey à plusieurs variables Preprint, University of Valladolid, Spain, 1992
  19. [Z2] M. A. Zurro The Abhyankar Jung theorem revisited J. Pure Appl. Algebra 90 (1993), 257-282
  20. [Z3] M. A. Zurro Series y funciones Gevrey en varias variables Ph.D. Thesis, University of Valladolid, Spain, 1994
  21. [Z4] M. A. Zurro Summability 'au plus petit terme' Studia Math. 113 (1995), 197-198
Banach Center Publications
1998/44/1
Export to PBN, Opens in new tab
Pages:
277-292
Main language of publication
English
Published
1998
Exact and natural sciences