This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real zeros of a real polynomial is bounded in terms of its number of non-zero coefficients, has an analog in $\mathbb{Q}_p$. In addition, it has an arithmetic variant from the theory of heights, for if an integer polynomial $f \in \mathbb{Z}[X]$ has a low enough (normalized) Mahler measure $\Big( \int_{S^1} \log{|f(z)| \, \frac{d\theta}{2\pi}} \Big) / \deg{f} \in \mathbb{R}^{\geq 0}$ - for example, if its $L^1$-norm is subexponential in the degree, - then a well-known therem of Bilu implies a bound $o(\deg{f})$ on the number of its (distinct) $\mathbb{R}$ or $\mathbb{Q}_p$ zeros. (Here, "low enough" means smaller than some $\varepsilon > 0$, and then the $o(\cdot)$ is in terms of this $\varepsilon$ being sufficiently small.)

A higher dimensional extension of Descartes's result is found in Khovanskii's theory of fewnomials, one of whose main thrusts (put broadly) is that a low arithmetic complexity in a system of real analytic equations exerts a tight control on the topology of its set of real solutions (a real analytic variety). As a typical illustration (sticking for simplicity to the algebraic situation), Descartes's result can be generalized as an upper bound on the Betti numbers - in particular, number of connected components - of a real hypersurface in $\{F = 0\} \subset \mathbb{RP}^k$ in terms of the number of monomials in the defining equation $F$.

I would like to ask:

  1. Are there any published results concerning $p$-adic topology variants of Khovanskii's "low complexity $\Rightarrow$ tame topology" theory? Or is this an unexplored topic? A specific example as above would be a control on the number and topology of the connected components of the Berkovich analytification in $\mathbb{P}_{/\mathbb{Q}_p}^{k,\mathrm{an}}$ of a hypersurface $\{F = 0\} \subset \mathbb{P}^k(\mathbb{Q}_p)$ defined by an equation $F$ over $\mathbb{Q}_p$ that contains a small number of monomials.

  2. Arithmetic variants over $\mathbb{Z}$, replacing the basic notion of arithmetic complexity (number of non-zero coefficients) by a canonical height in arithmetic geometry. On $\mathbb{G}_m^r$ for instance we may take a multivariate normalized logarithmic Mahler measure $m(F)/\deg{F} \in \mathbb{R}^{\geq 0}$ as above (replace the unit circle by the unit real $r$-torus), for an $F \in \mathbb{Z}[\mathbf{x}^{\pm 1}]$, and ask for the possibility that the smallness (in the sense of being close enough to zero) of this entropy-like complexity measure could exert any control on the topology of the real zero set $\{F = 0\} \cap \mathbb{G}_m^r(\mathbb{R})$.


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