An Isomorphism between Monoids of External Embeddings: - About Definability in Arithmetic - Mihai Prunescu *†‡ Abstract We use a new version of the Definability Theorem of Beth in order to unify classical Theorems of Yuri Matiyasevich and Jan Denef in one structural statement. We give similar forms for other important definability results from Arithmetic and Number Theory. A.M.S. Classification: Primary 03B99; Secondary 11D99. 1 Introduction It would be nice if there were some analogon of Galois Theory concerning the general definability, or even more the definability with syntactical restrictions (existential, diophantine). Efforts put by people into searching for concrete equations defining the subring of rational integers or other important predicates would be then replaced by structural thinking. We would not care anymore about the Pell equation, which does not seem to work on essentially other rings of algebraic integers as the rings solved in [Denef 2], [Denef-Lipshitz], by Thanases Pheidas in [Pheidas] and independently by Alexandra Shlapentokh in [Shlapentokh2]. We will present now a tentative to start such a program. Our results have more a philosophical than a practical nature. We will use our method in order to analyze already proved statements and to produce new information. We make the following convention: to speak about a T -definable or a T -diophantine relation over some polynomial ring R[T ] means to stress the fact that a constant T representing the element T occurs in the definition explicitly. We will use this notation only in some contexts where it is necessary. The known statements, we want to analyze are the classical Theorems of Yuri Matiyasevich and Jan Denef: Theorem 1.1 (Matiyasevich) An arbitrary relation over the ring Z of all rational integers is diophantine in Z if and only if it is recursively enumerable according to a recursive presentation of Z. Theorem 1.2 (Denef ) An arbitrary relation over the polynomial ring Z[T ] is T -diophantine in Z[T ] if and only if it is recursively enumerable according to a recursive presentation of Z[T ]. The recursive presentation of Z is trivial and does not appear explicitly in the different statements of the Theorem of Matiyasevich: it occurred here only for the sake of symmetry. Both proofs are * These results occurred in author’s Ph. D. dissertation at the University of Konstanz, Germany, ended in 1998. During this dissertation the author was partially supported by D.A.A.D. and by the University of Konstanz. The author kindly thanks his adviser, Prof. Alexander Prestel, for all support. The author also kindly thanks the anonymous referee, who did a great work for increasing the readability of this paper. † Institut f¨ ur Mathematik und Informatik, Universit¨at Greifswald, Germany. ‡ Institute of Mathematics of the Romanian Academy, Bucharest, Romania. 1