Open Questions over Number Fields

We hope that from our narrative it is clear that there is no shortage of open problems. In fact one could get an impression that for every question answered at least two open ones appear. There are many ways to organize these questions. We choose to divide them into two main collections: questions of definability and questions of Turing reducibility, including questions of decidability. Now the questions of definability can also be divided into many other categories. Given two rings R1 and R2 with fraction fields K1 and K2 being number fields, one can pose a number of definability problems.

  • 1. If R1 C R2 we can ask whether R1 has a Diophantine definition over R2. If R1 C R2 then one can ask whether R1 <Dioph R2 or whether R1 is Dioph- generated over R2. Diophantine generation is defined as follows. Let K be a number field containing both K1 and K 2 and let ш1,...,ш„ be any basis of K over K2. Now consider the question of existence of a Diophantine subset A C Rn2+l such that (a1,...,an, b) e A ^ b = 0 and R1 = {2П=1 f l(a1, ???,an, b) e A}. (For more details on Diophantine generation see [56].)
  • 2. More generally, we can ask whether R2 has a Diophantine model of R1 or a class Diophantine model of R1. A class Diophantine model corresponds to what model theorists call a Diophantine interpretation and is a map which establishes a correspondence between R1 and equivalence classes of elements of R2 under a Diophantine equivalence relation. Further, there should be a Diophantine description of the class of the products and sums. (For an example of a class Diophantine model see [60].)

Under any of these definability relations between R1 and R2 we can conclude that

Of course we can ask the weaker question of Turing reducibility directly about R1 and R2. Further, it would be interesting to see an example where we have Turing reducibility but no definability. As discussed above, we suspect that something like this may be true with respect to Q and Z but no version of the assertion claiming Turing reducibility without definability has been proved so far. One simple example which illustrates the difficulty of these questions is described below.

Question 3.1 Let R C Q be a big ring. Let p be a rational prime number not inverted in R, and let R = R[p]. In this case is R definable in any way over R (via Dioph-generation, Diophantine model, or Diophantine interpretation)? Is HTP(R) HTP(R)?

Note that if Z is Diophantine over R, than all of these questions can easily be answered in the affirmative. If we can define Z, then we can define powers of p and thus R. In some rings we can generate powers of some primes without defining Z but the general case remains quite vexing.

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