Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Success and challenges in determining the rational points on curves

Nils Bruin

Abstract: One of the central problems in number theory is to determine the rational solutions to a system of polynomial equations, or at least to decide if there are any such solutions. A completely general algorithmic solution to this problem seems presently completely out of reach and, in the line of the negative results for Hilbert's tenth problem, may well be fundamentally impossible.

When we restrict our attention to systems that describe algebraic curves, the outlook is much more positive. Indeed, there is a procedure that is in principle generally applicable and in many cases yields an answer that is guaranteed to be correct. The fact that we presently cannot guarantee that the procedure will always eventually yield an answer prevents us from claiming it an algorithm.

I will discuss this procedure, the ingredients it needs and the computational challenges that arise in providing those ingredients and in applying the procedure.

Files available: slides (PDF)