Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



First order differential closures of certain partially ordered fields

Author: Joseph E. Turcheck
Journal: Trans. Amer. Math. Soc. 195 (1974), 97-114
MSC: Primary 12H05; Secondary 02H15
MathSciNet review: 0340229
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: First order algebraic differential equations (a.d.e.'s) are considered in the setting of an abstract differential field with an abstract order relation, whose properties mirror those of the usual asymptotic dominance relations of analysis. An abstract existence theorem, for such equations, is proved by constructing an extension of both the differential field and the abstract order relation. As a consequence, a first order differential closure theorem, for those differential fields with order relations which we consider, is obtained. The closure theorem has corollaries which are important to the asymptotic theory of a.d.e.'s and have application to a.d.e.'s with coefficients meromorphic in a sector of the complex plane.

References [Enhancements On Off] (What's this?)

  • [1] S. Bank, On the instabliity theory of differential polynomials, Ann. Mat. Pura Appl. (4) 74 (1966), 83-111. MR 34 #4623. MR 0204785 (34:4623)
  • [2] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 33 #5416. MR 0197234 (33:5416)
  • [3] W. Strodt, On the algebraic closure of certain partially ordered fields, Trans. Amer. Math. Soc. 105 (1962), 229-250. MR 25 #3934. MR 0140514 (25:3934)
  • [4] W. Strodt, Graduated logarithmic fields and stability, Univ. of Wisconsin, U.S. Army MRC Tech. Summary Rep. No. 489, June, 1964.
  • [5] -, Asymptotic expansions and structure theorems, Preliminary report, Notices Amer. Math. Soc. 18 (1971), 370-371. Abstract #683-B27.
  • [6] W. Strodt and R. K. Wright, Asymptotic behavior of solutions and adjunction fields for nonlinear first order differential equations, Mem. Amer. Math. Soc. No. 109 (1971), 284 pp. MR 44 #1884. MR 0284660 (44:1884)
  • [7] J. Turcheck, Contributions to the abstract theory of asymptotic expansions, for solutions of algebraic differential equations, Dissertation, Columbia University, New York, N. Y., 1972.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 12H05, 02H15

Retrieve articles in all journals with MSC: 12H05, 02H15

Additional Information

Keywords: Abstract existence theorem, algebraic differential equation, asymptotically nonsingular, asymptotic order relation, attached, critical chain, critical monomial, critically unique, differential closure property, differential field extension, differential polynomial, finitely enveloped, graduated logarithmic field, logarithmic monomial, principal monomial, rank stable, stability of a differential polynomial, structure theorem, valuation
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society