When most people think about Galois Theory, they likely think of the classical case, in which one considers the roots of a given polynomial, and asks about the set of automorphisms of the complex plane which permute these roots. It is also possible that they think of the more general (and more modern) situation in which one takes any field and looks at the set of automorphisms of this field that fix a given subfield. A smaller group (though one this reviewer belongs to) thinks about algebraic geometry, where Galois theory has become a crucial part of understanding topics such as moduli spaces and schemes.

At the end of the nineteenth century, Picard and Vessiot began to consider a version of Galois theory for differential equations, and in the 1950s these ideas were formalized by various mathematicians starting with E. L. Kolchin, who called this new field “Differential Galois Theory.” This theory tries to understand solutions of differential equations by using the symmetries of the field generated by a complete set of solutions. Much work has been done in this area over the years, and it remains a vibrant field of research today.

To a certain strain of mathematician, once something is understood for differential equations it is natural to wonder what happens with their relatives the finite difference equations, and it is not surprising to learn that there has been significant work over the last few decades connecting Galois Theory to difference equations. For those unfamiliar with this area, the type of question one might ask is:

What are the sequences \(\{a_n\}\) so that \(a_{n+2}+n a_{n+1} + na_n=0\)?

One can easily check that one solution to the above question is given by \(a_{n} =(-1)^n k(n-2)\) for any constant \(k\), but it turns out there are others, and that Galois theory can help us understand the structure of these solutions. The new book *Galois Theories of Linear Difference Equations: An Introduction* collects notes from three courses taught at the CIMPA Research School in Colombia in 2012 exploring these connections. The authors only assume familiarity with algebra at the level of a first graduate course, although at times some experience with differential equations and analysis would be helpful, and throughout their notes they describe three different “Galoisian” approaches to studying linear difference equations.

The first article, “Algebraic and Algorithmic Aspects of Linear Difference Equations” is by Michael F. Singer, and covers some of the same material as his book with Marius van der Put. In particular he shows how one can start with a linear difference equation and associate to it a group of matrices, and then by studying the properties of this group one can determine properties of the solutions of the difference equation. In order to develop this theory, Singer dedicates a number of pages to introducing linear algebraic groups and algebraic varieties more generally. He also introduces the concept of difference rings and Picard-Vessiot extensions which allow us to understand the solutions to these difference equations. He also spends time looking at the computational aspects of the theory and he shows how these algorithms apply to several examples.

Another article is by Charlotte Hardouin, and acts in large part as an expository version of her 2008 article with Singer in Mathematische Annalen. In these notes, she introduces a number of topics in differential algebra in general and in the geometric side of the subject in particular. She goes on to develop what she refers to as parametrized Picard-Vessiot theory, which can be used to study the differential algebraic properties of solutions of linear differential equations. All of this theory builds to a proof of Holder’s theorem, which asserts that the Gamma function does not satisfy a polynomial differential equation over the complex rational functions. This result has been known since the late nineteenth century, but the modern proof is very exciting.

The third section of the book is by Jacques Sauloy and is entitled “Analytic Study of \(q\)-Difference Equations.” These \(q\)-difference equations arise (albeit with different notation and terminology) as early as the work of Euler, and play a critical role in understanding the Riemann Zeta function as well as the partition function (which recently made its big screen debut in *The Man Who Knew Infinity*) He also looks at the role that \(q\)-difference equations play as deformations of differential equations. Sauloy quickly moves the discussion from these questions to issues of monodromy and singularities. He is then able to define factorizations of certain operators which eventually leads to a normal form for difference equations.

As is often the case with volumes of lecture notes, the exposition throughout this book is somewhat inconsistent in tone and in the background it assumes. At times it is very elementary and methodical about its approach and at other times it breezes through details, leaving this reviewer wanting more. As such, a reader who wants to learn the material deeply would probably do better to read other books in the area that develop the material in a fuller way. But for someone who wants a flavor of the ways that ideas from Galois theory can be useful in understanding difference equations and are ok with having certain technical details swept under the rug, this book will be a good resource. I found the level of exposition to be good, particularly in Singer’s chapter, and there are a handful of exercises sprinkled through the text. This is a very interesting area of mathematics that more people should know about, and for the right person this book is a solid introduction.

Buy Now

Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His mathematical interests include number theory and algebraic geometry and the various versions of Galois theory that live in those worlds. He can be reached at dglass@gettysburg.edu.