Complex Cellular Structures

We introduce the notion of a \emph{complex cell}, a complexification of the cells/cylinders used in real tame geometry. Complex cells are equipped with a natural notion of holomorphic extension, and the hyperbolic geometry of a cell within its extension 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 sharpen 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. Combined with work of Burguet, Liao and Yang, our refined version establishes an optimal sharpening of these estimates for \emph{analytic} maps, 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 prove a sharpening of 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.