Mathematical Sciences: Diophantine Geometry in Positive Characteristics

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This award supports the research of Professor Voloch to work in the theory of Diophantine equations. The study of Diophantine equations, that is, the search for solutions in integers or rational numbers of polynomial equations, is a very classical topic in number theory. Much of the recent progress in the subject stems from insights obtained from the analogous problem when rational numbers are replaced by rational functions, the so-called function field case. This project will study this last case. Previous work of the author has led to general results of a qualitative nature. It is hoped in this project to obtain quantitative results, leading to effective bounds that allow one to actually find the solutions to the equations. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.