Mathematical Sciences: Some Problems in Algorithmic Number Theory and Diophantine Equations

这项拨款支持皮拉教授在数论方面的研究。他特别希望研究2属超椭圆曲线雅可比矩阵约简上的点数计数算法。他还打算对位于正方形内的曲线上的积分格点的数目得到改进的界限。这是数论领域的研究。数论从整数和诸如整数能被整数整除之类的问题开始。它是数学中最古老的领域之一，最初纯粹出于审美原因而追求它。然而，在过去的半个世纪里，它已经成为开发计算机科学新算法和电子学新纠错码等领域的重要工具。

This grant supports the research of Professor J. Pila to work in number theory. In particular, he wishes to work on algorithms for counting points on the reductions of Jacobians of hyperelliptic curves of genus 2. He also intends to work on obtaining improved bounds for the number of integral lattice points on curves lying inside squares. 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 areas such as developing new algorithms for computer science and new error correcting codes for electronics.