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Definability in reducts of algebraically closed fields

  • Gary A. Martin (a1)

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Let K be an algebraically closed field and let L be its canonical language; that is, L consists of all relations on K which are definable from addition, multiplication, and parameters from K. Two sublanguages L1 and L2 of L are definably equivalent if each relation in L1 can be defined by an L2-formula with parameters in K, and vice versa. The equivalence classes of sublanguages of L form a quotient lattice of the power set of L about which very little is known. We will not distinguish between a sublanguage and its equivalence class.

Let Lm denote the language of multiplication alone, and let La denote the language of addition alone. Let fK [X, Y] and consider the algebraic function defined by f (x, y) = 0 for x, yK. Let Lf denote the language consisting of the relation defined by f. The possibilities for LmLf are examined in §2, and the possibilities for LaLf are examined in §3. In fact the only comprehensive results known are under the additional hypothesis that f actually defines a rational function (i.e., when f is linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., when f is linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is that L covers Lm and that the only languages between La and L are expansions of La by scalar multiplications.

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[1]Hrushovski, Ehud, Contributions to stable model theory, Ph.D. thesis, University of California, Berkeley, California, 1986.
[2]Hungerford, Thomas W., Algebra, Holt, Rinehart and Winston, New York, 1974.
[3]Martin, Gary A., Two classification problems: rank one structures which coordinatize ℵ0-categorical, ℵ0-stable structures; reducts of algebraically closed fields, Ph.D. thesis, Rutgers University, New Brunswick, New Jersey, 1985.
[4]Poizat, Bruno, Une théorie de Galois imaginaire, this Journal, vol. 48 (1983), pp. 11511170.
[5]Zil′ber, B. I., Some problems concerning categorical theories, Bulletin de I’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 29 (1981), pp. 4749.

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