A trailer for p-adic analysis, second half: Mahler coefficients

In the previous post we defined ${p}$ -adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ . Then we give the famous proof of the Skolem-Mahler-Lech theorem using ${p}$ -adic analysis.

1. Digression on ${\mathbb C_p}$

Before I go on, I want to mention that ${\mathbb Q_p}$ is not algebraically closed. So, we can take its algebraic closure ${\overline{\mathbb Q_p}}$ — but this field is now no longer complete (in the topological sense). However, we can then take the completion of this space to obtain ${\mathbb C_p}$ . In general, completing an algebraically closed field remains algebraically closed, and so there is a larger space ${\mathbb C_p}$ which is algebraically closed and complete. This space is called the ${p}$ -adic complex numbers.

We won’t need ${\mathbb C_p}$ at all in what follows, so you can forget everything you just read.

2. Mahler coefficients: a description of continuous functions on ${\mathbb Z_p}$

One of the big surprises of ${p}$ -adic analysis is that we can concretely describe all continuous functions ${\mathbb Z_p \rightarrow \mathbb Q_p}$ . They are given by a basis of functions

$\displaystyle \binom xn \overset{\mathrm{def}}{=} \frac{x(x-1) \dots (x-(n-1))}{n!}$

in the following way.

Theorem 1 (Mahler; see Schikhof Theorem 51.1 and Exercise 51.B)

Let ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ be continuous, and define

$\displaystyle a_n = \sum_{k=0}^n \binom nk (-1)^{n-k} f(n). \ \ \ \ \ (1)$

Then ${\lim_n a_n = 0}$ and

$\displaystyle f(x) = \sum_{n \ge 0} a_n \binom xn.$

Conversely, if ${a_n}$ is any sequence converging to zero, then ${f(x) = \sum_{n \ge 0} a_n \binom xn}$ defines a continuous function satisfying (1).

The ${a_i}$ are called the Mahler coefficients of ${f}$ .

Exercise 2

Last post we proved that if ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ is continuous and ${f(n) = (-1)^n}$ for every ${n \in \mathbb Z_{\ge 0}}$ then ${p = 2}$ . Re-prove this using Mahler’s theorem, and this time show conversely that a unique such ${f}$ exists when ${p=2}$ .

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”.

$\displaystyle \begin{aligned} a_0 &= f(0) \\ a_1 &= f(1) - f(0) \\ a_2 &= f(2) - 2f(1) - f(0) \\ a_3 &= f(3) - 3f(2) + 3f(1) - f(0). \end{aligned}$

Thus one can think of ${a_n \rightarrow 0}$ as saying that the values of ${f(0)}$ , ${f(1)}$ , \dots behave like a polynomial modulo ${p^e}$ for every ${e \ge 0}$ . Amusingly, this fact was used on a USA TST in 2011:

Exercise 3 (USA TST 2011/3)

Let ${p}$ be a prime. We say that a sequence of integers ${\{z_n\}_{n=0}^\infty}$ is a ${p}$ -pod if for each ${e \geq 0}$ , there is an ${N \geq 0}$ such that whenever ${m \geq N}$ , ${p^e}$ divides the sum

$\displaystyle \sum_{k=0}^m (-1)^k \binom mk z_k.$

Prove that if both sequences ${\{x_n\}_{n=0}^\infty}$ and ${\{y_n\}_{n=0}^\infty}$ are ${p}$ -pods, then the sequence ${\{x_n y_n\}_{n=0}^\infty}$ is a ${p}$ -pod.

3. Analytic functions

We say that a function ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ is analytic if it has a power series expansion

$\displaystyle \sum_{n \ge 0} c_n x^n \quad c_n \in \mathbb Q_p \qquad\text{ converging for } x \in \mathbb Z_p.$

As before there is a characterization in terms of the Mahler coefficients:

Theorem 4 (Schikhof Theorem 54.4)

The function ${f(x) = \sum_{n \ge 0} a_n \binom xn}$ is analytic if and only if

$\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{n!} = 0.$

Just as holomorphic functions have finitely many zeros, we have the following result on analytic functions on ${\mathbb Z_p}$ .

Theorem 5 (Strassmann’s theorem)

Let ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ be analytic. Then ${f}$ has finitely many zeros.

4. Skolem-Mahler-Lech

We close off with an application of the analyticity results above.

Theorem 6 (Skolem-Mahler-Lech)

Let ${(x_i)_{i \ge 0}}$ be an integral linear recurrence. Then the zero set of ${x_i}$ is eventually periodic.

Proof: According to the theory of linear recurrences, there exists a matrix ${A}$ such that we can write ${x_i}$ as a dot product

$\displaystyle x_i = \left< A^i u, v \right>.$

Let ${p}$ be a prime not dividing ${\det A}$ . Let ${T}$ be an integer such that ${A^T \equiv \mathbf{1} \pmod p}$ .

Fix any ${0 \le r < N}$ . We will prove that either all the terms

$\displaystyle f(n) = x_{nT+r} \qquad n = 0, 1, \dots$

are zero, or at most finitely many of them are. This will conclude the proof.

Let ${A^T = \mathbf{1} + pB}$ for some integer matrix ${B}$ . We have

$\displaystyle \begin{aligned} f(n) &= \left< A^{nT+r} u, v \right> = \left< (\mathbf1 + pB)^n A^r u, v \right> \\ &= \sum_{k \ge 0} \binom nk \cdot p^n \left< B^n A^r u, v \right> \\ &= \sum_{k \ge 0} a_n \binom nk \qquad \text{ where } a_n = p^n \left< B^n A^r u, v \right> \in p^n \mathbb Z. \end{aligned}$

Thus we have written ${f}$ in Mahler form. Initially, we define ${f \colon \mathbb Z_{\ge 0} \rightarrow \mathbb Z}$ , but by Mahler’s theorem (since ${\lim_n a_n = 0}$ ) it follows that ${f}$ extends to a function ${f \colon \mathbb Z_p \rightarrow \mathbb Q_p}$ . Also, we can check that ${\lim_n \frac{a_n}{n!} = 0}$ hence ${f}$ is even analytic.