1
$\begingroup$

Let $f: \mathbb Z_{\ge 0} \to \mathbb C_p$ be any function. My understanding is that Mahler's theorem says that $f$ extends to a continuous function $f: \mathbb Z_p \to \mathbb C_p$ if and only if the Mahler coefficients \begin{align*} d_0 &= f(0) \\ d_1 &= f(1)-f(0) \\ d_2 &= f(2)-2f(1)+f(0) \\ &\vdots \\ d_n &= \sum_k (-1)^{n-k} \binom nk f(k) \end{align*} tend to zero in the $p$-adic sense. By finite differences, this is equivalent to the existence of a polynomial $P_e \in \mathbb Q[x]$, taking only integer values on $\mathbb Z$, for which $f(n) \equiv P_e(n) \pmod{p^e}$ holds for $n \ge 0$. (Please correct me if this is wrong!) In either case, we have the explicit formula $$f(x) = \sum_k d_k \binom xk \qquad x \in \mathbb Z_p.$$

In reading proofs of Skolem-Mahler-Lech theorem (e.g. here), I have the following additional questions.

  1. Are there conditions (in terms of either $d_n$ or $P_e$) on when the extended function $f : \mathbb Z_p \to \mathbb C_p$ is analytic?

  2. Assuming $f$ isn't identically zero, are there conditions on which $f$ can have only finitely many zeros? (In particular, the blog post linked seems to claim that analytic is sufficient, but as $\mathbb Z_p$ is quite disconnected I'm confused why this is true.)

$\endgroup$
  • 2
    $\begingroup$ For your last question, analytic power series have finitely many zeroes follows from the Weierstrass preparation theorem. See math.stackexchange.com/a/1417765/68188 $\endgroup$ – Asvin Oct 14 '17 at 16:25
  • 2
    $\begingroup$ Analytic in the sense that it has a power series expansion? Or Krasner analytic? $\endgroup$ – EFinat-S Oct 14 '17 at 17:00
  • $\begingroup$ Anyway, I'm pretty sure this appears in Schikhof's or Robert's books. This kind of results are due to Yvette Amice, if I'm correct. If I have time later I'll search. $\endgroup$ – EFinat-S Oct 14 '17 at 17:39
4
$\begingroup$

For your first question: Theorem 54.4 (page 166) in Schikhof's "Ultrametric Calculus" says that $f$ is analytic if and only if

$$\lim_{n\to\infty}\frac{d_n}{n!}=0.$$

Here, analytic means that $f$ has a power series expansion which converges for all $|x|\le1$ (in $\mathbb{Q}_p$, hence in $\mathbb{C}_p$).

Actually, in his book Schikhof gives characterizations in terms of Mahler coefficients of a lot of interesting spaces of $p$-adic functions. Many of them are due to Yvette Amice.

If your meaning of analytic is "Krasner analytic", you may look at Robert's "A course in $p$-adic analysis", section 4.3, page 344.

For your second question, as Asvin points out, a nonzero power series on the closed unit ball has only finitely many zeros there ("Strassman theorem", see Robert, page 306).

$\endgroup$
  • 1
    $\begingroup$ Stress that you mean at the end the closed unit ball $|x| \leq 1$. A nonzero power series converging on the open unit ball $|x| < 1$ could have infinitely many zeros there, such as the $p$-adic logarithm. $\endgroup$ – KConrad Oct 15 '17 at 1:27
  • $\begingroup$ Of course, "unit ball" without open/closed is always confunsing. Thanks. $\endgroup$ – EFinat-S Oct 15 '17 at 1:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.