BOUNDS ON TRANSFER PRINCIPLES FOR ALBEGRAICALLY CLOSED AND COMPLETE DISCRETELY VALUED FIELDS* I. INTRODUCTION We study the replacement of quantifiers in statements about algebraically closed fields or complete discretely valued fields with residue class field Z p and p large compared to the length of the statement, by quantifiers which range over finite sets of algebraic numbers. Since these algebraic numbers may be effectively determined and manipulated, this gives decision procedures in these cases. It also yields the following bounds on well known transfer principles. Let x ** v = x y an( j x ** y** z = x **(y**z). THEOREM 1: If S is a first order statement in the language with plus, minus, times, and constant symbols 0, 1, 10, 11, 100, . .. whose representation is m characters long and p 2 ** 2** 2** (3m) then S holds in the theory of algebraically closed fields of characteristic p if and only if it holds in the theory of algebraically closed fields of characteristic zero. THEOREM 2: If S is a statement of length m in the language of theorem 1 along with a constant symbol which represents a uniformizing parameter and p 2 * * 2 * * 2 * * 2 * * 2 * * l l * * m then S holds in the p-adic field Q p if and only if it holds in the field of formal power series with finitely many negative terms over the field with p elements Zp((t)). Theorem 2 bounds the exceptions to the Artin conjecture as: THEOREM 3: For every natural number d any form of degree d in more thaa d **2 variables has a nontrivial zero in Qp provided p 2 **2 **2 **2 **2 ** 11 ** d ** (4d). We begin with an abstract discussion of our method of limitation of quantifiers in a structure K and illustrate it in the additive group of rational numbers, the additive ordered group of integers, and the ordered field of real numbers. Sets of linear functions, for the additive theories, or polynomials, for the field, delimit equivalence classes in Kn where all the atomic subformulae of a quantifier-free formula have constant truth value Quantification corresponds to projecting these classes to Kn , . . . , K , K and the projected classes are delimited by simultaneously solving the linear equations or taking discriminants and resultants of the poly- nomials. For the rational numbers the equivalence classes are determined from the set of * Received by the editors 4 February 1977 revised 29 September 1977. This work is a revised version of the author's doctoral dissertation at Princeton University, 1976, which was supported by a National Science Foundation graduate fellowship. 1
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