10 problems

Positivity of the rank. There is $c>0$ such that for all sufficiently large $x$, $N(x)\ge c\cdot x$. This conjecture is motivated by work of Helfgott and, together with a parity st…

Let $L/K$ be an extension of number fields. They are connected by a diophantine chain if there is an integer $n\geq 0$ and a tower of number fields … such that, for every $i$ with…

Sea $F$ un campo de números y sea $O_F$ su anillo de enteros. Un subconjunto de $O_F$ es diofantino si existe una ecuación polinómica con coeficientes en $O_F$ cuyos valores de las…

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

Let $K$ be a global field and let $\mathscr{G}$ be a finite set of field generators. Consider $K$ as a structure over $\mathscr{L}=\mathscr{L}_a\cup\mathscr{G}$. Global-field rank…

A real number $\alpha$ is left-listable if $L(\alpha)=\{q\in\mathbb{Q}:q<\alpha\}$ is listable. Let $\mathscr{D}$ be the set of left-Diophantine numbers and let $\Lambda$ be the se…

Let $\mathscr{D}\subseteq\mathbb{R}$ be the set of left-Diophantine numbers. Field conjecture. The set $\mathscr{D}$ is a field. This is a weaker consequence of the algebraicity co…

A real number $x\in\mathbb{R}$ is left-Diophantine if it is the supremum of a Diophantine subset of $\mathbb{Q}$. Let $\mathscr{D}$ be the set of left-Diophantine numbers, and let…

Let $K\subset L$ be number fields. Let $d\in\mathcal O_K$ be nonzero and satisfy the hypotheses of Theorem--in particular, the hypotheses referred to as -- in the source. For the c…

The rank-preserving elliptic curves conjecture. There exists such a $d$ for which $(0,d)$ has infinite order and generates $E(L)\otimes\mathbb Q$. This would provide a rank-preserv…