NSF/CBMS Regional Conference in the Mathematical Sciences -- Solving Polynomial Equations

数学中最基本的问题之一是多项式方程的求解。这样的系统在应用数学中无处不在，出现在机器人、编码理论、优化、数学生物学、计算机视觉、统计学和许多其他领域。当然，多项式系统的研究本身就是一个美丽的话题，并且是代数几何的主题，代数几何传统上被认为是纯数学中深奥而困难的主题。近年来，几何计算的显式算法和实用软件的爆炸性发展彻底改变了这一领域，使许多以前无法解决的问题变得容易处理，并为实验和猜想提供了肥沃的土壤。这些算法也引起了科学家、工程师和应用数学家对代数几何的惊人兴趣。该领域使用的工具涵盖了数学的各个领域，从数值方法和微分方程到代数几何再到组合学。这些讲座的目的是概述最近的结果和“艺术状态”；并讨论这些结果所带来的大量开放问题，并提出新的探索方向。

One of the most fundamental problems in mathematics is that of solving polynomial equations. Such systems are ubiquitous in applied mathematics, arising in robotics, coding theory, optimization, mathematical biology, computer vision, statistics, and numerous other areas. Of course, the study of polynomialsystems is a beautiful topic in its own right and is the subject of algebraic geometry, which istraditionally regarded as a deep and difficult subject in pure mathematics.In recent years, an explosive development in explicit algorithms and practical software for geometric calculations has revolutionized the area, making many formerly inaccessable problems tractable, and providing a fertile ground for experimentation and conjecture. These algorithms have also generated a surprising interest in algebraic geometry among scientists, engineers, and applied mathematicians. Thetools of the trade employed in this field span the spectrum of mathematics, ranging from numericalmethods and differential equations to algebraic geometry to combinatorics. The purpose of these lectures is toprovide an overview of recent results and the ``state of the art''; and to discuss the wealth of open questions raised by these results and suggest new directions ripe for exploration.