5 problems

Hilbert irreducibility conjecture. The polynomial $f$ remains irreducible for infinitely many integer specializations of $s$.

Let $K$ be a number field, and let $X\rightarrow \mathbb{A}^1_K$ be a non-trivial geometrically irreducible branched cover of curves over $K$. Weak Diversity conjecture. There exis…

Let $X\rightarrow \mathbb{A}^1_{\mathbb{Q}}$ be a geometrically irreducible branched cover of curves over $\mathbb{Q}$, such that not all of its branch points are $\mathbb{Q}$-rati…

Let $X$ be a curve over $\mathbb{Q}$, let $tmin \mathbb{Q}(X)$ be a non-constant $\mathbb{Q}$-rational function of degree at least $2$, and let $P_1,ldots,P_N$ be the points in the…

Let $X$ be a curve over $\mathbb{Q}$ and let $tmin \mathbb{Q}(X)$ be a non-constant $\mathbb{Q}$-rational function of degree at least $2$. For the points $P_1,ldots,P_N$ in the abo…