Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of Q

As Easy as $\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields

Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\mathbb Q)$. We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r \leq 1$ we construct such a ring $R = Z[\frac1p : p \in S]$ with $S$ a set of primes of lower density $r$. We …

Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields

We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t>1, and t nonnegative computable real numbers delta_1,..., delta_t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S_1,..., S_t of densities delta_1,..., delta_t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O_{K,S_i}. We also …

Descent on Elliptic Curves and Hilbert's Tenth Problem

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.

Baire category theory and Hilbert's Tenth Problem inside $\mathbb{Q}$

For a ring R, Hilbert's Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP($\mathbb{Q}$), whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP($\mathbb{Q}$) computes C if and only if the subrings R for which HTP(R) computes C form a nonmeager class. Similar results hold for 1-reducibility, for …

Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields

We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophant…

椭圆曲线平均意义上的丢番图稳定性

Diophantine stability for elliptic curves on average

本文证明正密度的秩1椭圆曲线在素数处具有丢番图稳定性，并探讨其对希尔伯特第十问题的影响。