Uncomputably large integral points on algebraic plane curves?

We show that the decidability of an amplification of Hilbert’s Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prefixes ∃∀∃ and ∃∃∀∃ are generically decidable. This means, taking the former prefix as an example, that we give a precise geometric classification of those polynomials f ∈Z[v, x, y] for which the question ∃v∈N such that ∀x∈N ∃y∈N with f(v, x, y)=0? may be undecidable, and we show that this set of polynomials is quite small in a rigourous sense. (The decidability of ∃∀∃ was previously an open question.) The analogous result for the prefix ∃∃∀∃ is even stronger. We thus obtain a connection between the decidability of certain Diophantine problems, height bounds for points on curves, and the geometry of certain complex surfaces and 3-folds.