10 problems

Let $\mathrm{HTP}(2)$ denote the problem of deciding whether an arbitrary polynomial in $\mathbb{Z}[x,y]$ has a root in $\mathbb{N}^2$. Its analogue over $\mathbb{Z}^2$ asks whethe…

Use the source's positive-integer Diophantine prefixes $\exists\forall\exists$ and $\exists\exists$, the function $\mathbf{Big}_{\mathbb{N},2}$ measuring the maximal size of positi…

For $f\in\mathbb{Z}[x_1,x_2]$, let $\mathbf{Big}_{\mathbb{N},2}(f)$ be the supremum of the maximum coordinate among the positive integral points on the plane curve $f=0$, with the…

Let $L_0$ be a field and let $L_1$ be a nontrivial finitely generated extension of $L_0$. Suppose there exists a valuation $v$ on $L_1$ with residue field $L_0$. Weak resolution of…

Large-prime existence conjecture. For all sufficiently large primes $p$, there exist an elliptic curve $E/\mathbb{Q}$ and an imaginary quadratic field $K_0$ such that the triple…

The conjecture on the final hypothesis. Martin Davis conjectured that this remaining number-theoretic hypothesis would be solved by a clever young Russian mathematician in the near…

Let $D\subseteq \mathbb{C}(z)$ be a Diophantine set. For each integer $n\geq 0$, write $\mathbb{C}[z]_n=\mathbb{C}^{n+1}$ for the polynomials of degree at most $n$. Kollár's conjec…

Baker–Matijasevič–Robinson conjecture. The theory $\exists^3$ is undecidable over $\mathbb{N}$.

Let $S\subseteq\mathbb{N}$ be a simple set, meaning that $\mathbb{N}\setminus S$ is infinite and contains no infinite recursively enumerable subset. Let $U\subseteq\mathbb{Q}$ be t…

Let $\mathbb{Z}$ denote the ring of integers and $\mathbb{Q}$ the field of rational numbers. A subset of $\mathbb{Q}$ is Diophantine if it is defined by the existence of solutions…