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I found on the Internet some ways to provide a graphical representation of the $p$-adic integers or numbers (e.g., these illustrations of Heiko Knospe). They all exploit the fact that $p$-adic integers have a tree-like structure, given by their representation as a series $\sum_{i=0}^\infty a_i p^i$, with $a_i \in \{0,1,\dots,p-1\}$. Thus one gets some tree-like or fractal-like images.

However, I wonder, how can one give a graphical representation of a $p$-adic function? Say the $p$-adic exponential or the $p$-adic logarithm.

The issue is that, using these tree-like representations, for every point of the domain of the function, one should "plot" a branch of a tree, and I do not see how to do that graphically.

Thanks for any suggestion.

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  • $\begingroup$ One may make a much cruder plot of the behavior of, for instance, the logarithm, than you were thinking of, namely: The “basepoint” of the logarithm function $w=\log z$ is at $z=1$, so you may look in the neighborhood of $1$ by plotting $v((z)$ against $v(\log(1+z))$. That’s the additive valuation $v$, according to which $v(p^m)=m$. (Think of this as the negative of the (real) logarithm of $|z|$.) It’s additive in the sense that $v(zz')=v(z)+v(z')$. If you’d like more dtailed information, I can write up an answer, but it would require a graphic input, which would be somethingi of a pain. $\endgroup$
    – Lubin
    Commented Jul 19, 2022 at 18:13
  • $\begingroup$ Have you tried ploting with $\sum_{i=0}^\infty a_i p^{-i}$ replacing the $p$-adic number $\sum_{i=0}^\infty a_i p^{i}$ instead? $\endgroup$
    – Somos
    Commented Jul 19, 2022 at 21:46
  • $\begingroup$ @Lubin It seems to me that you are suggesting to plot the p-adic valuation of the function, instead of the value of the function. This would be a very rough approximation, since the p-adic valuation is discrete. $\endgroup$
    – Perry
    Commented Jul 20, 2022 at 6:57
  • $\begingroup$ @Somos the problem with such map is that it is not a bijection. Indeed, the representation $\sum_{i=-\infty}^\infty a_i p^i$ of $p$-adic numbers is unique, while the representation $\sum_{i=-\infty}^\infty a_i p^{-i}$ of real numbers is not. $\endgroup$
    – Perry
    Commented Jul 20, 2022 at 7:05
  • $\begingroup$ The non-uniqueness of the map is true theoretically, but for the purposes of graphical representation it is good enough in my opinion. Have you tried this approach yet? $\endgroup$
    – Somos
    Commented Jul 20, 2022 at 10:38

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