Definability and decidability problems in number theory

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5. Hilbert’s Tenth Problem for Subrings of $\mathbb Q$

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Problem 5.2.
[Hector Pasten] Is there a diophantine subset $X$ of $\mathbb Q$ such that $\sup\{x\mid x\in X\}$ is transcendental?
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Problem 5.3.
[Florian Pop]
Let $X/\mathbb Q$ be a variety, and let $\{Y_a\}_{a\in A}$ be a set of uniformly definable subsets of $X$ .

Is $\overline{Y_a(\mathbb Q)} = Y_a(\mathbb R)$ ?
Does this imply that $\mathbb Z$ is not diophantine in $\mathbb Q$ ?
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Problem 5.6.
What is the structure of HTP for big rings in $\mathbb Q$ under $\leq_T$ ? (Here, “big rings" mean rings where infinitely many primes are inverted.)

Cite this as: AimPL: Definability and decidability problems in number theory, available at http://aimpl.org/definedecide.