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The group of automorphisms of a differential algebraic function field

  • Michihiko Matsuda (a1)

Abstract

Consider a one-dimensional differential algebraic function field K over an algebraically closed ordinary differential field k of characteristic 0. We shall prove the following theorem:

Suppose that the group of all automorphisms of K over k is infinite. Then, K is either a differential elliptic function field over k or K = k(ν) with ν′ = ξ or ν′ = ην, where ξ, η ϵ k.

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References

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[1] Forsyth, A. R., A treatise on differential equations, 6th ed., Macmillan and Co., London, 1933.
[2] Hurwitz, A., Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403442.
[3] Iwasawa, K., Theory of algebraic function fields (in Japanese), Iwanami Shoten, Tokyo, 1952.
[4] Kolchin, E. R., Galois theory of differential fields, Amer. J. Math. 75 (1953), 753824.
[5] Matsuda, M., Liouville’s theorem on a transcendental equation logy = y/x , J. Math. Kyoto Univ. 16 (1976), 545554.
[6] Matsuda, M., Algebraic differential equations of the first order free from parametric singularities from the differential-algebraic standpoint, J. Math. Soc. Japan, 30 (1978), 447455.
[7] Rosenlicht, M., On the explicit solvability of certain transcendental equations, Publ. Math. Inst. HES., no. 36 (1969), 1522.
[8] Rosenlicht, M., An analogue of L’Hospital’s rule, Proc. Amer. Math. Soc. 37 (1973), 369373.
[9] Shafarevich, I. R., Basic algebraic geometry (in Russian), Nauka, Moscow, 1972: Translation (in English), Springer, Berlin, 1974.
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The group of automorphisms of a differential algebraic function field

  • Michihiko Matsuda (a1)

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