First-order definability of affine Campana points in the projective line over a number field

Abstract

We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible $S$, are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial.