Let me say that I am a big fan of Iain Banks. I've read everything he's written, and loved about 90% of it. (The other 10% is just too violent and disgusting for me.) This is one of the books that I enjoyed. The war, the characters, and especially the subplot about artificial intelligence are all great. However, from a mathematical point of view, I was quite disappointed.

I will elaborate below, but might give away some of the plot twists in doing so. Therefore, if you're planning to read the book anyway, you should probably stop reading this now.

According to a well known "conspiracy theory", there is a hidden network of wormholes controlled by the dwellers. A list of coordinates is already in wide circulation which is supposedly part of the information needed, but people have been unable to work out exactly what to do with the information. The footnote in The Algebraist leads people to further speculate that some sort of mathematical step is necessary...perhaps a "transform".

Now, I like the use of the word "transform" here. Like the title ("algebraist" is the correct term for a mathematician specializing in algebra) it is a technical mathematical term that seems to be used correctly. Perhaps Banks was thinking of something complicated like a Fourier transform which is used in signal processing to break waves down into their harmonic components, but just about any formula that could turn the list of numbers into another useful list of numbers could be called a "transform".

The part that disappoints me is his later use of the term "equation" and what the answer eventually turns out to be. In the end, Fassin finds the clue which he is surprised to find is an "equation" rather than a "transform". That already bugs me, because a transform could well be presented in the form of an equation. But, even worse, it turns out that although Banks repeatedly calls the thing an "equation", it is not an equation at all. In particular, there is no "equality" involved in the thing at all. He seems to be (mis)using "equation" to mean just some mathematical expression.

Then, when the mathematical expression is finally worked out, it turns out to be something amazingly simple. This is okay, I guess. It is supposed to be ironic that the answer was so simple all along. However, I both did not believe that this simple answer would really help you to work out where the wormholes were located and did not believe that nobody would ever have thought of it even without finding this clue!

So, in conclusion, I'd say that this is a nice science fiction novel, but not a particularly good work of mathematical fiction. Although the title and quest are both mathematical, very little math appears in the book and the little that does was somewhat disappointing.