The investigator will work on two projects connected with number theory and arithmetic geometry. The first project is to study generalizations of Hilbert's Tenth Problem. The problem in its original form asked for an algorithm to decide whether an arbitrary multivariable polynomial equation with integer coefficients has an integer solution. In 1970 Matiyasevich proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. This motivated studying analogues of this problem by considering equations and solutions in other commutative rings. The biggest open problem in the area is Hilbert's Tenth Problem over the rational numbers. The PI has proved the undecidability of Hilbert's Tenth Problem for various function fields. These generalizations have used tools from arithmetic geometry, such as the study of rational points on elliptic curves. One research goal is to extend these results and prove undecidability of Hilbert's Tenth Problem for the function fields for which the problem is still unresolved. The biggest open problems are function fields of one variable over an algebraically closed field. Another goal is to explore Hilbert's Tenth Problem for various subrings of number fields. The second project is to study several problems that deal with computational aspects of curves and their Jacobians. Elliptic curves and, more generally, Jacobians of curves of small genus have many applications to cryptography, and the second project focuses on these applications. One goal of the second project is to explore curves of small genus and work on constructing curves that are suitable for cryptographic purposes. The PI will also work on applications of pairings to cryptography.
Both projects involve studying the solutions to multivariable polynomial equations. Looking for solutions to such equations over the integers or rational numbers has a long history that goes back to ancient Greece. For the first project the investigator will study the fundamental question of whether it is possible to find a procedure that determines whether an arbitrary multivariable polynomial equation has a solution in a given number system. The second project focuses on computational aspects of certain special classes of equations that have applications to cryptography. For these applications one usually looks for solutions to these equations over finite fields.
