Integrally Hilbertian rings and the polynomial Schinzel hypothesis

The Hilbert specialization property is one of the few general and powerful tools in Arithmetic Geometry, allowing parameters to be specialized without modifying the algebraic structure. A fundamental instance is the Hilbert Irreducibility Theorem, which states that for any irreducible polynomial \(P \in \mathbb{Q}[T, Y]\) of positive degree in \(Y\), infinitely many specializations \(m \in \mathbb{Q}\) yield irreducible polynomials \(P(m,Y)\). One of the main motivations for Hilbert was the Inverse Galois Problem, as this theorem enables the construction of extensions of \(\mathbb{Q}\) with prescribed Galois group from extensions of \(\mathbb{Q}(T)\).
Bodin, Dèbes, König, and Najib extended this theorem to integral settings, replacing \(\mathbb{Q}\) by \(\mathbb{Z}\) or the ring of integers \(\mathcal{O}\) of number fields with class number \(1\). In joint work with Pierre Dèbes, we remove the class number restriction and apply these results to a polynomial version of the Schinzel Hypothesis. In the classical setting, this hypothesis predicts that under natural local conditions, finite families of irreducible integer polynomials simultaneously take prime values infinitely often—covering, in particular, conjectures such as the twin prime conjecture and the infinitude of primes of the form \(m^2 + 1\).