In the previous post we defined -adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions . Then we give the famous proof of the Skolem-Mahler-Lech theorem using -adic analysis.
1. Digression on
Before I go on, I want to mention that is not algebraically closed. So, we can take its algebraic closure — but this field is now no longer complete (in the topological sense). However, we can then take the completion of this space to obtain . In general, completing an algebraically closed field remains algebraically closed, and so there is a larger space which is algebraically closed and complete. This space is called the -adic complex numbers.
We won’t need at all in what follows, so you can forget everything you just read.
2. Mahler coefficients: a description of continuous functions on
One of the big surprises of -adic analysis is that we can concretely describe all continuous functions . They are given by a basis of functions
in the following way.
The are called the Mahler coefficients of .
You’ll note that these are the same finite differences that one uses on polynomials in high school math contests, which is why they are also called “Mahler differences”.
Thus one can think of as saying that the values of , , \dots behave like a polynomial modulo for every . Amusingly, this fact was used on a USA TST in 2011:
3. Analytic functions
We say that a function is analytic if it has a power series expansion
As before there is a characterization in terms of the Mahler coefficients:
Just as holomorphic functions have finitely many zeros, we have the following result on analytic functions on .
We close off with an application of the analyticity results above.
Proof: According to the theory of linear recurrences, there exists a matrix such that we can write as a dot product
Let be a prime not dividing . Let be an integer such that .
Fix any . We will prove that either all the terms
are zero, or at most finitely many of them are. This will conclude the proof.
Let for some integer matrix . We have
Thus we have written in Mahler form. Initially, we define , but by Mahler’s theorem (since ) it follows that extends to a function . Also, we can check that hence is even analytic.
Thus by Strassman’s theorem, is either identically zero, or else it has finitely many zeros, as desired.