Extensions of Hilbert's Tenth Problem and Computational Aspects of Arithmetic Geometry

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.