  We introduce the notion of a complex cell, a complexification of the
cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex
cell $\mathcal{C}$ we define its holomorphic extension
$\mathcal{C}\subset\mathcal{C}^\delta$, which is again a complex cell. The
hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^\delta$ provides the
class of complex cells with a rich geometric function theory absent in the real
case. We use this to prove a complex analog of the cellular decomposition
theorem of real tame geometry. In the algebraic case we show that the
complexity of such decompositions depends polynomially on the degrees of the
equations involved.
  Using this theory, we refine the Yomdin-Gromov algebraic lemma on
$C^r$-smooth parametrizations of semialgebraic sets: we show that the number of
$C^r$ charts can be taken to be polynomial in the smoothness order $r$ and in
the complexity of the set. The algebraic lemma was initially invented in the
work of Yomdin and Gromov to produce estimates for the topological entropy of
$C^\infty$ maps. For analytic maps our refined version, combined with work of
Burguet, Liao and Yang, establishes an optimal refinement of these estimates in
the form of tight bounds on the tail entropy and volume growth. This settles a
conjecture of Yomdin who proved the same result in dimension two in 1991. A
self-contained proof of these estimates using the refined algebraic lemma is
given in an appendix by Yomdin.
  The algebraic lemma has more recently been used in the study of rational
points on algebraic and transcendental varieties. We use the theory of complex
cells in these two directions. In the algebraic context we refine a result of
Heath-Brown on interpolating rational points in algebraic varieties. In the
transcendental context we prove an interpolation result for (unrestricted)
logarithmic images of subanalytic sets.
