Bjorn Poonen

Abstract:   Hilbert's Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients would decide whether there exists a solution in integers. In 1970, Matijasevic, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous problem with Z replaced by Q is still unknown, and there is not even agreement among experts as to what the answer should be. Attempts to prove a negative answer have come into conflict with conjectures of Mazur on the ``topology of rational points.''