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Jean-Pierre Serre (whose papers are, of course, always worth reading) recently posted a lovely lecture on the arXiv entitled “How to use finite fields for problems concerning infinite fields”. In it, he describes several ways in which algebraic statements over fields of zero characteristic, such as ${{\mathbb C}}$, can be deduced from their positive characteristic counterparts such as ${F_{p^m}}$, despite the fact that there is no non-trivial field homomorphism between the two types of fields. In particular finitary tools, including such basic concepts as cardinality, can now be deployed to establish infinitary results. This leads to some simple and elegant proofs of non-trivial algebraic results which are not easy to establish by other means.

One deduction of this type is based on the idea that positive characteristic fields can partially model zero characteristic fields, and proceeds like this: if a certain algebraic statement failed over (say) ${{\mathbb C}}$, then there should be a “finitary algebraic” obstruction that “witnesses” this failure over ${{\mathbb C}}$. Because this obstruction is both finitary and algebraic, it must also be definable in some (large) finite characteristic, thus leading to a comparable failure over a finite characteristic field. Taking contrapositives, one obtains the claim.

Algebra is definitely not my own field of expertise, but it is interesting to note that similar themes have also come up in my own area of additive combinatorics (and more generally arithmetic combinatorics), because the combinatorics of addition and multiplication on finite sets is definitely of a “finitary algebraic” nature. For instance, a recent paper of Vu, Wood, and Wood establishes a finitary “Freiman-type” homomorphism from (finite subsets of) the complex numbers to large finite fields that allows them to pull back many results in arithmetic combinatorics in finite fields (e.g. the sum-product theorem) to the complex plane. (Van Vu and I also used a similar trick to control the singularity property of random sign matrices by first mapping them into finite fields in which cardinality arguments became available.) And I have a particular fondness for correspondences between finitary and infinitary mathematics; the correspondence Serre discusses is slightly different from the one I discuss for instance in here or here, although there seems to be a common theme of “compactness” (or of model theory) tying these correspondences together.

As one of his examples, Serre cites one of my own favourite results in algebra, discovered independently by Ax and by Grothendieck (and then rediscovered many times since). Here is a special case of that theorem:

Theorem 1 (Ax-Grothendieck theorem, special case) Let ${P: {\mathbb C}^n \rightarrow {\mathbb C}^n}$ be a polynomial map from a complex vector space to itself. If ${P}$ is injective, then ${P}$ is bijective.

The full version of the theorem allows one to replace ${{\mathbb C}^n}$ by an algebraic variety ${X}$ over any algebraically closed field, and for ${P}$ to be an morphism from the algebraic variety ${X}$ to itself, but for simplicity I will just discuss the above special case. This theorem is not at all obvious; it is not too difficult (see Lemma 4 below) to show that the Jacobian of ${P}$ is non-degenerate, but this does not come close to solving the problem since one would then be faced with the notorious Jacobian conjecture. Also, the claim fails if “polynomial” is replaced by “holomorphic”, due to the existence of Fatou-Bieberbach domains.

In this post I would like to give the proof of Theorem 1 based on finite fields as mentioned by Serre, as well as another elegant proof of Rudin that combines algebra with some elementary complex variable methods. (There are several other proofs of this theorem and its generalisations, for instance a topological proof by Borel, which I will not discuss here.)

Update, March 8: Some corrections to the finite field proof. Thanks to Matthias Aschenbrenner also for clarifying the relationship with Tarski’s theorem and some further references.