Baire category theory and Hilbert’s Tenth Problem inside Q

As Easy as Q: Hilbert’s Tenth Problem for Subrings of the Rationals and Number Fields

Hilbert’s Tenth Problem over the rationals 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 ⊆ Q having the property that Hilbert’s Tenth Problem for R, denoted HTP(R), is Turing equivalent to HTP(Q). We are able to put several additional constraints on the rings R that we construct. Given any co...

Measure theory and Hilbert’s Tenth Problem inside 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. When R = Z, it is known that the jump Z′ is Turing-reducible to HTP(Z). We consider computability of HTP(R) for subrings R of the rationals. Applying measure theory to these subrings, which naturally form a measure space, relates their sets HTP(R) to the set HTP(Q),...

Hilbert’s Tenth Problem and Mazur’s Conjecture for Large Subrings of Q

We give the first examples of infinite sets of primes S such that Hilbert’s Tenth Problem over Z[S−1] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E′ over Z[S−1] such that the topological closure of E′(Z[S−1]) in E′(R) has infinitely many connected components.

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. 1. Hilbert’s Tenth P...

Notes on Hilbert’s Tenth Problem over Q