Functions satisfying elementary relations
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- by Michael F. Singer
- Trans. Amer. Math. Soc. 227 (1977), 185-206
- DOI: https://doi.org/10.1090/S0002-9947-1977-0568865-2
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Abstract:
In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form $R = 0$ where R is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations.References
- James Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252–268. MR 277482, DOI 10.2307/1970774
- Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. MR 0042164, DOI 10.1090/surv/006
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
- Joseph Fels Ritt, Integration in Finite Terms. Liouville’s Theory of Elementary Methods, Columbia University Press, New York, N. Y., 1948. MR 0024949, DOI 10.7312/ritt91596
- J. F. Ritt, On the integrals of elementary functions, Trans. Amer. Math. Soc. 25 (1923), no. 2, 211–222. MR 1501240, DOI 10.1090/S0002-9947-1923-1501240-7
- Maxwell Rosenlicht, Liouville’s theorem on functions with elementary integrals, Pacific J. Math. 24 (1968), 153–161. MR 223346, DOI 10.2140/pjm.1968.24.153
- Maxwell Rosenlicht, On the explicit solvability of certain transcendental equations, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 15–22. MR 258808, DOI 10.1007/BF02684595 —, On Liouville’s theory of elementary functions, Pacific J. Math. (to appear). M. Singer, Functions satisfying elementary relations, Thesis, Univ. of California, Berkeley, 1974.
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 227 (1977), 185-206
- MSC: Primary 12H05
- DOI: https://doi.org/10.1090/S0002-9947-1977-0568865-2
- MathSciNet review: 0568865