The Yomdin-Gromov algebraic lemma states that a semialgebraic subset of the unit cube can be parameterized by finitely many C^r-smooth charts of unit C^r norms, and that the number of these charts can be chosen to depend only on the degrees of the polynomials defining the set. This result is the main tool in Yomdin's proof of Shub's entropy conjecture for C^\infty-smooth maps. It has also been used more recently in the study of rational points on algebraic and transcendental varieties. Getting sharp control over the number of charts is the key to open problems in both the dynamical and diophantine directions.
I will discuss a new complex analytic approach to the algebraic lemma. First, I will show that the appearance of C^k (rather than holomorphic) charts is due to obstructions related to the hyperbolic metric on the complex disc. I will then introduce "complex cells" complexifying the cellular decompositions of semialgebraic geometry, and show that their hyperbolic structure gives rise to a rich geometric function theory. Finally I will describe how using this function theory one can prove an essentially sharp version of the algebraic lemma. In particular this resolves a conjecture of Yomdin from 1991 on the topological entropy of real-analytic maps. The results are joint work with D. Novikov.