
## 1. Computability

1. #### Problem 1.1.

[Russell Miller] Given $f\in \mathbb Z[x_1,\dots, x_n]$, define the following sets. $$A(f) = \{\text{subrings of }\mathbb Q \text{ where f has a solution}\}$$ $$\mathcal C(f) = \text{The interior of the complement of A(f)}$$ $$B(f) =\text{The boundary of the complement of A(f)} %= \text{the boundary set"}$$
1. Can the Lebesgue measure of $B(f)$ be positive?
2. If so, what is the maximal complexity of the measure of $B(f)$?
3. Is $\mathcal C(f)$ always a finite union of basic open sets?

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