Abstract
In this paper we survey some generalizations of formal Laurent power series to several indeterminates and we expound some of the fundamental logical results concerning fields of generalized power series. In connection with the above, we also present the notions of saturated model and of ultraproduct.
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References
J. Ax and S. Kochen, Diophantine problems over local fields I, Amer. J. Math. 187(1965)605–630.
J. Ax and S. Kochen, Diophantine problems over local fields III: Decidable fields, Ann. Math. 83(1966)437–456.
J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in:Model Theory of Algebra and Arithmetic, Lecture Notes in Mathematics 834 (Springer, Berlin, 1980) pp. 1–9.
C.C. Chang and H.J. Keisler,Model Theory (North-Holland, New York, 1973).
G. Cherlin,Model Theoretic Algebra, Lecture Notes in Mathematics 521 (Springer, Berlin, 1976).
P.J. Cohen, Decision procedures for real andp-adic fields, Commun. Pure Appl. Math. 22(1969)131–151.
F. Delon, Indécidabilité de la théorie des anneaux de séries formelles à plusieurs indéterminées, Fundamenta Math. 62(1981)215–229.
G.A. Elliott and P. Ribenboim, Fields of generalized power series, Archiv. d. Math. 54(1990)365–371.
M.J. Greenberg,Lectures on Forms in Many Variables (Benjamin, New York, 1969).
W. Hodges,Model Theory (Cambridge University Press, Cambridge, 1993).
C.U. Jensen and H. Lenzing,Model Theoretic Algebra (Gordon and Breach, New York, 1989).
I. Kaplansky, Maximal fields with valuations, Duke Math. J. 9(1942)303–321.
S. Kochen, The model theory of local fields,Logic Conf. Kiel 1974, Lecture Notes in Mathematics 499 (Springer, Berlin, 1975).
W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167(1932)160–196.
F.V. Kuhlmann, Henselian function fields, Dissertation, Heidelberg (1989).
A.I. Malcev, On the embedding of group algebras, Dokl. Akad. Nauk SSSR 60(1948)1499–1501 (in Russian).
B.H. Neumann, On ordered division rings, Trans. AMS 66(1949)202–252.
P. Ribenboim,Théorie des Valuations (Les Presses de l'Université de Montréal, Montréal, 1964).
A. Robinson, Elementary embeddings of field of power series, J. Number Th. 2(1970)237–247.
G. Sabbagh, Aspects logiques de la pureté dans les modules, C.R.A.S. Paris A271(1970)909–912.
L. van de Dries, Model theory of fields, Thesis, Utrecht (1978).
R.J. Walker,Algebraic Curves (Dover Publ., New York, 1962).
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Delon, F. Formal power series. Ann Math Artif Intell 16, 59–73 (1996). https://doi.org/10.1007/BF02127794
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DOI: https://doi.org/10.1007/BF02127794