Relating decision and search algorithms for rational points on curves of higher genus

For affine plane curves defined over the rationals of genus at least two, we show that a decision algorithm for the existence of solutions also yields a search algorithm for all solutions.Comment: to be published in Archive for Mathematical Logi

Existence of rational points on smooth projective varieties

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Chatelet surfaces such that exactly one of the surfaces fails to have a k-point.Comment: 11 page

Relating decision and search algorithms for rational points on curves of higher genus.

In the study of rational solutions to polynomial equations in two-variables, we show that an algorithmic solution to the decision problem (existence of solutions) enables one to construct a search algorithm for all solutions