Exponentially Algebraically Closed Fields

幂运算是继加法和乘法之后最基本的数学运算。它出现在描述指数增长和衰减时，出现在描述统计学正态分布的高斯曲线中，出现在物理学中出现的许多基本微分方程的解中。在复数上，幂运算也捕获正弦和余弦函数，并且对于更普遍地建模周期行为是必不可少的。尽管指数函数无处不在，但关于指数函数的一些最基本的代数问题仍未得到解答。具体来说，给定一个使用加法、乘法和幂运算的多变量方程组，通常不知道该方程组是否有复数的解。对于不取幂的相应问题，也就是多项式方程组，答案是肯定的。它可以归结为所谓的代数基本定理和希尔伯特的零定理，自19世纪末以来一直为人所知。本课题的目的是证明代数基本定理的指数类比，即证明复数是指数代数闭的（EAC）。用现代术语来说，代数基本定理指出复数域是代数封闭的，希尔伯特的Nullstellensatz则描述了多项式方程组在代数封闭域中是否有解。后一个定理的指数类比是由Zilber给出的，有时被称为Zilber的Nullstellensatz。因此，证明复数的EAC性质将解决指数方程组在复数中是否有解的问题。这个项目将向几个方向进行。在系统只包含一个方程的特殊情况下，期望的结果是已知的，在某些条件下，两个或多个方程也是已知的。在一个方向上，我们将进一步推动现有的分析技术，旨在获得两个，三个或更多方程的完整结果。解析技术包括寻找近似解，然后改进近似并证明它们收敛于精确解。在第二个方向上，我们将开发新的技术，利用代数几何和同伦理论的思想来解决同样的问题。这种方法包括考虑解如何随着方程的变化而连续变化，并得出结论，即使不知道解的确切位置，解也必须实际存在。在第三个方向上，我们将发现沿几何线的方程组的新分类，它将指导我们使用其他方法。第四个方向是使用我们开发的技术来解决其他相关问题，例如解决涉及除幂以外的操作的方程组。在许多情况下，所考虑的方程组的解可以在复平面上或通过动画图形化地说明。这个项目的另一个方面是开发这样的插图，并用它们向数学研究界以外的观众解释研究。

Exponentiation is the most fundamental mathematical operation after addition and multiplication. It arises when describing exponential growth and decay, in the Gaussian curves describing normal distributions for statistics, and in solutions to many of the basic differential equations which arise in physics. On the complex numbers, exponentiation also captures the sine and cosine functions, and is essential to model periodic behaviour more generally.Despite its ubiquity, some of the most basic algebraic questions about the exponential function remain unanswered. Specifically, given a system of equations in several variables using the operations of addition, multiplication and exponentiation, in general it is not known if that system has a solution in the complex numbers. The answer to the corresponding question without exponentiation, that is, for systems of polynomial equations, is yes. It boils down to the so-called Fundamental Theorem of Algebra and Hilbert's Nullstellensatz, and has been known since the end of the 19th century.The aim of this project is to prove the exponential analogue of the Fundamental Theorem of Algebra, that is, to show that the complex numbers are Exponentially Algebraically Closed (EAC). In modern terms, The Fundamental Theorem of Algebra states that the field of complex numbers is algebraically closed, and Hilbert's Nullstellensatz then characterizes whether or not a system of polynomial equations has solutions in an algebraically closed field. The exponential analogue of the latter theorem was given by Zilber, and is sometimes called Zilber's Nullstellensatz. Thus proving the EAC property for the complex numbers would solve the problem of whether a system of exponential equations has a solution in the complex numbers. The project will proceed in several directions. The desired result is known in the special case when the system contains only one equation, and also under certain conditions for two or more equations. In one direction we will push existing techniques from analysis further, aiming to get the complete result for two, three, or more equations. Analytic techniques involve finding approximate solutions, and then improving the approximations and showing that they converge to exact solutions. In a second direction we will develop new techniques using ideas from algebraic geometry and homotopy theory to attack the same problems.This approach involves considering how solutions must vary continuously as the equations vary, and concluding that the solutions must actually exist even without knowing exactly where they are. In a third direction we will find a new classification of the systems of equations along geometric lines, which will guide our use of the other methods. A fourth direction is to use the techniques we develop to tackle other related problems, such as solving systems of equations which involve operations other than exponentiation.In many cases, the solutions of the systems of equations under consideration can be graphically illustrated in the complex plane or via animations. A further aspect of this project is to develop such illustrations and use them to explain the research to an audience outside the mathematics research community.