The profinite Schinzel hypothesis H over S-integers

Let $D$ be the ring of $S$-integers in a global field, let $f_1,\dots,f_d$ be pairwise non-associate separable irreducible polynomials over $D$, and let $f=\prod_{i=1}^d f_i$. Assume that no prime ideal $\mathfrak p$ divides $f_1(n)\cdots f_d(n)$ for every $n\in D$.

Extended Schinzel hypothesis H. If

$$f(\widehat D)\cap\widehat D^*\ne\varnothing,$$

then the set

$$\mathbf f(D)\cap\mathcal P(D)^d$$

is infinite.

This is the profinite formulation of the infinitude part of Schinzel’s hypothesis H: the unit-intersection condition expresses the absence of local obstructions. It is open in general and includes simultaneous prime values of several polynomials.