ABSTRACT We establish the following bounds for well known transfer principles. Let x**y = x^ and 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 **ll**m then S holds in the p-adic field Qp 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 than d **2 variables has a nontrivial zero in Qp provided p 2 ** 2 **2 ** 2 ** 2 ** 11 **d ** (4d). In each case, the bound is established by showing how the statement's quantifiers may be replaced by quantifiers which range over finite sets of algebraic numbers. This also proves the decidability of the statements. AMS (MOS) subject classifications (1970). Primary 02E10, 02H15 secondary 10C20, 12L05, 12B05. Key words and phrases: algebraically closed field," Artin conjecture, complete discretely valued field, limitation of quantifiers, p-adic, transfer principle. Library of Congress Cataloging in Publication Data W ^ | Brown, Scott Shorey, 1951- Bounds on transfer principles for algebraically closed and complete discretely valued fields . (Memoirs of the American Mathematical Society no. 20*0 A revision of the author's thesis , Princeton University, 1976. Bibliography: p. Includes index. 1. Algebraic fields . 2. Transfer functions. 3. p-adic numbers. I . Title . I I . Series American Mathematical Soceity. Memoirs no. 20^. QAJ.A57 no. 20l+ [QA2^7] 510'. 8s [512'. 7k] ISBN 0-8218-220^-7 78-9121 1978 © American Mathematical Society ii
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