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

研究人员将从事两个与数论和算术几何有关的项目。第一个项目是研究希尔伯特第十问题的推广。该问题的原始形式要求一种算法来确定任意整数系数的多变量多项式方程是否有整数解。1970年，Matiyasevich证明了不存在这样的算法，即希尔伯特的第十问题是不可判定的。这激发了通过考虑其他交换环中的方程和解来研究这个问题的类似问题。这个领域最大的开放问题是希尔伯特关于有理数的第十问题。π证明了各种函数域的希尔伯特第十问题的不可判定性。这些推广使用了算术几何的工具，例如对椭圆曲线上有理点的研究。一个研究目标是扩展这些结果，并为问题尚未解决的函数域提供希尔伯特第十问题的不可判定性。最大的开放问题是代数闭域上的单变量函数域。另一个目标是探索希尔伯特第十问题的各种子数域。第二个项目是研究处理曲线及其雅可比矩阵的计算方面的几个问题。椭圆曲线和更一般地说，小属曲线的雅可比矩阵在密码学中有许多应用，第二个项目着重于这些应用。第二个项目的目标之一是探索小属曲线，并致力于构建适合密码学目的的曲线。PI还将研究配对在密码学中的应用。这两个项目都涉及研究多变量多项式方程的解。在整数或有理数上寻找这样的方程的解有很长的历史，可以追溯到古希腊。对于第一个项目，研究者将研究一个基本问题，即是否有可能找到一个程序来确定任意多变量多项式方程在给定的数系中是否有解。第二个项目侧重于某些特殊类型的方程的计算方面，这些方程在密码学中有应用。对于这些应用，人们通常在有限域上寻找这些方程的解。

The investigator will work on two projects connected with numbertheory and arithmetic geometry. The first project is to studygeneralizations of Hilbert's Tenth Problem. The problem in itsoriginal form asked for an algorithm to decide whether an arbitrarymultivariable polynomial equation with integer coefficients has aninteger solution. In 1970 Matiyasevich proved that no such algorithmexists, i.e. Hilbert's Tenth Problem is undecidable. This motivatedstudying analogues of this problem by considering equations andsolutions in other commutative rings. The biggest open problem in thearea is Hilbert's Tenth Problem over the rational numbers. The PI hasproved the undecidability of Hilbert's Tenth Problem for variousfunction fields. These generalizations have used tools fromarithmetic geometry, such as the study of rational points on ellipticcurves. One research goal is to extend these results and proveundecidability of Hilbert's Tenth Problem for the function fields for whichthe problem is still unresolved. The biggest open problems arefunction fields of one variable over an algebraically closed field.Another goal is to explore Hilbert's Tenth Problem for varioussubrings of number fields. The second project is to study severalproblems that deal with computational aspects of curves and theirJacobians. Elliptic curves and, more generally, Jacobians of curvesof small genus have many applications to cryptography, and the secondproject focuses on these applications. One goal of the second projectis to explore curves of small genus and work on constructing curvesthat are suitable for cryptographic purposes. The PI will also work onapplications of pairings to cryptography.Both projects involve studying the solutions to multivariablepolynomial equations. Looking for solutions to such equations over theintegers or rational numbers has a long history that goes back toancient Greece. For the first project the investigator will study thefundamental question of whether it is possible to find a procedurethat determines whether an arbitrary multivariable polynomial equationhas a solution in a given number system. The second project focuseson computational aspects of certain special classes of equations thathave applications to cryptography. For these applications one usuallylooks for solutions to these equations over finite fields.