Diophantine equations look very innocent at first glance. Say we want to find rational numbers \(x\), \(y\), \(z\) such that \(x^4+y^4+z^4=1\). Seems like a reasonable question, an attackable problem. How would we approach it?

A first idea might be to clear denominators, so that we can search for integer solutions instead, but at the cost of introducing an extra variable: if \(d\) is the least common denominator, our equation now looks like \(a^4+b^4+c^4=d^4\) with \(a,b,c,d\) nonzero integers.

We now have two obvious moves to consider. The first is straightforward: try small values for \(d\) and try to determine whether \(d^4\) can be written as a sum of three fourth powers. Set a computer to work on this and wait… and wait…

Oh, well. Perhaps we should try the second move: prove there isn’t any solution. Maybe there is a sign obstruction? If the initial equation had been \(x^4+y^4+z^4=-1\) we would have seen at once that it had no solution, after all. But in our case there’s no such problem.

The next obvious thing to do is this: if there are solutions with \(a,b,c,d\in\mathbb{Z}\), then there are also solutions to any congruence \(a^4+b^4+c^4\equiv d^4\pmod{m}\). If we can find an \(m\) for which we can show there are no solutions, we will have shown there are no integer solutions.

Alas, the straightforward ideas both fail for this example, taken from Karim Belabas’s introduction to the book under review. There are solutions, so trying to find a congruence obstruction is not going to work. On the other hand, the smallest solution is \[ 95800^4+217519^4+414560^4=422180^4,\] and no “dumb search” is ever going to find that. (In fact, even *checking* that it is a solution is non-trivial without a decent piece of mathematical software.) That solution was found by a very smart combination of theory and search: a lot of very fancy mathematical theory was deployed to restrict the search space enough so that a computer could have a chance of finding a solution. This despite the fact that one can prove that the set of rational triples \((x,y,z)\) such that \(x^4+y^4+z^4=1\) is *dense* in the set of real solutions.

And that is what *Explicit Methods in Number Theory* is about: the use of high-powered theorems to attack the problem of finding rational solutions to polynomial equations. The spirit of these articles, based on short courses taught during a topical trimester at the Institut Henri Poincaré, is well captured by the abstract of the first article:

The emphasis is not on developing algorithms for practical use, but on viewing the quest for polynomial-time algorithms as a challenge of our structural understanding.

That’s a great point: how can we claim we understand something if we can’t actually obtain any explicit results or do any specific computations? Of course, “practical use” is likely to follow in due time from increased understanding.

The articles are written by well-known masters in this field. They cover the main areas where we understand a least a part of the story: finite fields, elliptic curves, connections to modular forms, etc. The result is a book that anyone with an interest in arithmetical algebraic geometry will want to study.

Fernando Q. Gouvêa is someone “with an interest in arithmetical algebraic geometry.”