Explicit and computational approaches to arithmetic geometry

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A large part of the mathematical sciences is concerned with developing methods for finding solutions to various types of equations. The simplest kind of equation is the polynomial equation, where the variables are combined using just addition and multiplication. This is the oldest type of equation that was ever studied. One calls such an equation a Diophantine equation if one restricts to solutions that are natural numbers, integers, or rational numbers. Equations like this come up in many practical situations, such as in resonance problems, translating election results into seat distributions, and various questions about discrete configurations. It is generally a very hard problem to decide if a polynomial equation has an integer valued solution. In fact, in response to a challenge that Hilbert set in 1900, it has been proved that it is fundamentally impossible to create a general algorithm (computer program) that decides if there is an integral solution to a given polynomial equation. In contrast, the answer to the similar problem for rational solutions is unknown. For a special class of equations called curves, recent work has resulted in a method that can often decide whether there are any rational solutions in practice. Heuristic arguments suggest that these methods might always work and for a special class of curves, called hyperelliptic curves, practical experiments confirm this. This research program aims to improve our understanding of the rational solutions of polynomial equations. It will develop new and improve existing methods for deciding if there are any solutions at all and to determine all solutions if they exist. It will particularly support efforts to ensure that computational methods get extended beyond the special class of hyperelliptic curves.