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 prefix There Exists For All There Exists is generically decidable. This means that we give a precise geometric classification of those polynomials f is an element of Z[v, x, y] for which the question There Exists v is an element of N such that For All x is an element of N There Exists y is an element of N with f(v, x, y) = 0? may be undecidable, and we show that this set of polynomials is quite small in a rigorous sense. (The decidability of There Exists For All There Exists was previously an open question.) We also show that if integral points on curves can be bounded effectively, then There Exists There Exists For All There Exists is generically decidable as well. 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. (C) 2000 Elsevier Science B.V. All rights reserved.
integral points algebraic plane curves decidability diophantine complexity algebraic surfaces computability