Alexandra Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), no. 1, 117--134.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... extension of a totally real number field, then there exists a set of places S of K of Dirichlet density arbitrarily close to 1 [K : Q] 1 such that if is the subring of elements of K that are integral at all places outside S, then Hilbert s Tenth Problem over has a negative answer [Shl97] [Shl00], Shl02] But for K = Q, this gives nothing beyond Matijasevic s Theorem. More generally, we prove the following: Theorem 1.3. There exist disjoint recursive sets of primes T 1 and T 2 , both of natural density 0, such that for any set S of primes containing T 1 and disjoint from T 2 , the ....

Alexandra Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), no. 1, 117--134.

....extensions of Q, their totally complex extensions of degree 2, fields with exactly one pair of complex conjugate embeddings, some fields of degree 4, and some totally real infinite extensions of Q. For more details concerning these results see [5] 7] 8] 19] 21] 23] 24] 25] 27] and [29]. However, not much progress has been made towards resolving the Diophantine problem of Q. Further, one of the consequences of a series of conjectures by Barry Mazur and Colliot Thelene, Swinnerton Dyer and Skorobogatov is that Z does not have a Diophantine definition over Q, and thus one would ....

....that we have no definability result of this sort for Q. We do have some definability results for non trivial totally real extensions of Q and their totally complex extensions of degree 2, but even there the highest achievable density is one minus one over the degree of the field minus #. See [27] [29] and [23] for more details. 2 Some Density Results for Function Fields. This section makes extensive use of various properties of Dirichlet density. In particular, we use Chebotarev s Density Theorem to prove some of our results. The reader is referred to Chapter 5 of [9] for an exposition of ....

Alexandra Shlapentokh. Defining integrality at prime sets of high density in number fields. Duke Mathematical Journal, 101(1):117--134, 2000.

....rational primes WQ below W di#ers from SQ by a set contained in a set of Dirichlet density less than # and such that Z has a Diophantine definition over OK,W . Again this will imply that Hilbert s Tenth Problem is undecidable in OK,W . The proofs of these theorems can be found in [23] 20] and [18]. In this paper we consider the integral closures of the rings of W integers in some totally real infinite extensions of rational numbers. In general much less is known about the existential definability and decidability in the infinite extensions of Q than in the finite extensions. Due to ....

....below. The following lemma explains why. 3.1 Lemma. Let x # OMLE be a solution to the following system of equations. # NMLE ML (x) 1 NMLE EM (x) 1 (3.1) Then x 2hKLE # ELK. Further such a solution exists. 5 Proof. This lemma is similar to lemmas which can be found in [19] or [18] and deal with finite extensions. To prove the lemma in our case we had to change a few details. First we observe that no prime of WM p M , where p M is the M prime above p, splits in the extension ME M . Indeed, suppose t M # WM splits in the extension ME M . Since ME M is cyclic of ....

Alexandra Shlapentokh. Defining integrality at prime sets of high density in number fields. Duke Mathematical Journal. To appear.

....of Q, their totally complex extensions of degree 2, fields with exactly one pair of complex conjugate embeddings, some fields of degree 4, and some totally real infinite extensions of Q. For more details concerning these results see [8] 10] 11] 21] 24] 25] 26] 27] 29] and [30]. However, not much progress has been made towards resolving the Diophantine problem of Q. Further, one of the consequences of a series of conjectures by Barry Mazur and Colliot Thelene, Swinnerton Dyer and Skorobogatov is that Z does not have a Diophantine definition over Q, and thus one would ....

Alexandra Shlapentokh. Defining integrality at prime sets of high density in number fields. Duke Mathematical Journal, 101(1):117--134, 2000.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC