Exponentially Algebraically Closed Fields

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.

This project is in pure mathematics, and its most direct impact will be within mathematics, as explained in the section on Academic Beneficiaries and in the Case for Support. In this section we concentrate on the other impact the project will have.

In contrast to much research in pure mathematics, the main subjects of this project -- complex numbers and the exponential function -- are subjects which the general public can engage with enthusiastically. We will take advantage of that by developing workshops and talks around the subject matter of the project aimed at different general audiences, from school pupils, undergraduates, to the general public. These presentations will be given as part of the UEA outreach programme and also the public engagement programme of the School of Mathematics at UEA. They will take place in schools, as caf\'e conversations, and as part of other events such as the Norwich Science Festival. The workshops and talks will feature computer visualisations of the systems of equations we are solving, to give an insight into the geometry at play.


One difficulty with modern mathematics is that different branches necessarily become very specialised and hard for anyone
not working in that branch to understand. There can be a danger of isolated branches becoming somewhat separated from the rest of mathematics and from the world outside research mathematics. Preliminary work on this project has already involved discussions between the PI, working in model theory, and other mathematicians whose specialities are in numerical analysis, fluid dynamics, number theory, and set theory. These interactions will continue through the project, and so the project will build and strengthen connections between different areas of mathematics at a deep level. Such projects ensure that mathematicians from different areas are involved in explaining their work to a broader mathematical audience, which in turn helps them understand how to explain it to an audience outside mathematics. Thus projects such as this which build connections inside mathematics also have an indirect impact on the understanding and appreciation of mathematics in the wider population.