Mathematical Conversations

In 1900 Hilbert asked for a decision procedure to determine solvability of polynomial equations over the integers. Seventy years later, Y. Matiyasevich showed that this problem is unsolvable, building on earlier work of M. Davis, H. Putnam and J. Robinson. On the other hand, although we know that rational numbers form an undecidable field (J. Robinson, 1948), the decision problem for solvability of polynomial equations over the rational numbers remains open. In this talk I will discuss why the most commonly used methods to show undecidability can be expected to fail in this case. Several conjectures will be discussed, some of them false and some of them with no evidence.