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Plug each (n + l)-tuple into j(P) to see whether the result is j(0), starting with the first (n + l)-tuple not processed ¯ ¯ − 1). Record the first l elements of the first in the calculation of φ(0), . . , φ(m ¯ (n + l)-tuple satisfying the condition above as the value of φ(m). 5. 6. Let R1 , R2 be two integral domains such that R2 ≤Dioph R1 . Let F1 , F2 be the fraction fields of R1 and R2 respectively, and let j : F1 → N be a weak presentation of F1 . 2, and for any such extension J we have that J (R2 ) ≤e J (R1 ).

Bn ) = i, j Bi, j,k ai b j , and note that Hk (T1 , . . , T2n ) depends on the basis only. Then n n n bjωj = ai ωi i=1 j=1 Hk (a1 , . . , an , b1 , . . , bn )ωk . 2, for all k = 1, . . , n we know that j(Hk ) is extendable to a recursive function. ,Cn (m 2 )) J (×)(m 1 , m 2 ) = pi . i=1 Next we move to the translation of division. Since we have shown that the J -translation of multiplication in K is a restriction of a recursive function, it is enough to show that the J -translation of finding the multiplicative inverse is a restriction of a recursive function.

The term “Diophantine model” belongs to Gunther Cornelissen who introduced the terminology and the notion, in a slightly different form, in [5]. 4 Integrality at finitely many primes and divisibility of order at infinitely many primes In this chapter we will continue with the task of describing the known Diophantine classes of the rings of W-integers of global fields. We will start with horizontal problems. The question which we will partially answer here is the following. Does the Diophantine class of a ring of W-integers change if we add to or remove from W finitely many primes?

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