Uncomputably large integral points on algebraic plane curves? Academic Article uri icon


  • 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 is generically decidable. This means that we give a precise geometric classification of those polynomials f [v,x,y] for which the question v such that x y 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 was previously an open question.) We also show that if integral points on curves can be bounded effectively, then 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. 2000 Elsevier Science B.V. All rights reserved.

published proceedings

  • Theoretical Computer Science

author list (cited authors)

  • Rojas, J. M.

citation count

  • 4

complete list of authors

  • Rojas, J Maurice

publication date

  • January 2000