Diophantine Properties of Big and Small Rings

Before trying to answer the questions above, one should observe that the big and small rings share many Diophantine properties with the integers:

Proposition 3.6 1. The set of non-zero elements of a big or a small ring is Dio- phantine over the ring.

  • 2. A finite system ofpolynomial equations over a big ora small ring can be rewritten effectively as a single polynomial equation such that the solution set for the system is the same as the solution set for the equation.
  • 3. The set of non-negative elements of a big or a small ring R is Diophantine over R: a small modification of the Lagrange argument is required to accommodate possible denominators

It turned out that we already knew everything we needed to know about small rings from the work of J. Robinson (see [39]). In particular from her work on the first-order definability of integers over Q one can deduce the following theorem and corollaries.

Theorem 3.3 (J. Robinson) For every p, the ring Rp = {x e Q|x = m, m, n e Z, n > 0, p I n} has a Diophantine definition over Q.

This theorem of J. Robinson will play a role in many other results, as we will see below. In particular, now using Proposition 3.3 and Parts 1 and 2 of Proposition 3.6 we get the following corollaries.

Corollary 3.2 Z has a Diophantine definition over any small subring of Q.

Proof To see that J. Robinson’s theorem implies this corollary, let R be any big or small ring and observe that, since Q is not algebraically closed, Rp П R is Diophantine over R by Proposition 3.3 and Part 1 of Proposition 3.6. Let gR,p be the resulting Diophantine definition.

Now let R be a small ring with pi,..., pr being all the primes allowed in the denominators of its elements. Let gR (t, u) = 0 be the polynomial equation equivalent to the system

and observe that gR (t, u) is a Diophantine definition of R П Rpi П ••• П Rpr. (Existence of g(t, u) is guaranteed by Part 2 of Proposition 3.6.) Suppose t is an element of this intersection. Since t e R, the only primes that can divide the (reduced) denominator of t are pi,..., pr. However, being an element of Rpi, i = i,...,r implies that pi for all values of the index does not divide the denominator. Therefore, no prime can divide the denominator of t, and hence t is an integer. At the same time, trivially, Z c R n Rpi П ••• П Rpr. Thus, Z has a Diophantine definition over R. ?

Given that Z has a Diophantine definition over R, we apply Lemma 3.2 to conclude the following.

Corollary 3.3 HTP is unsolvable over all small subrings of Q.

Over big rings the questions turned out to be far more difficult.

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