Complex cellular structures

Abstract

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For δ ∈ (0, 1) and a complex cell C we define its holomorphic extension C ⊂ C δ , which is again a complex cell. The hyperbolic geometry of C within C δ 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 rsmooth 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 ∞ 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.

Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

“…In this paper, we built on the methods of [4] to further refine their result in Theorem 3.12. It is weaker than the result of [1] for subanalytic sets but holds for the larger class of power-subanalytic sets.…”

“…The dependence on r has recently been investigated in [1] by Binyamini and Novikov, who construct a C r -parameterization consisting of cr m maps for subanalytic sets. Here, c is a constant that depends on X.…”

“…The main obstacle to obtain a better result (e.g. cr m maps as in [1]), is the power substitution using r m to ensure that the walls of the cell are mild up to order r too. Thus it would be interesting to prove a stronger version of Theorem 3.11 that has better walls.…”

The dimension growth conjecture, polynomial in the degree and without logarithmic factors

“…By a careful elaboration of the argument from [11,Remark 2.3] and an explicit but otherwise classical projection argument, we find the following improvement of bounds by Bombieri-Pila [3,Theorem 5] and later sharpenings by Pila [21], [22], Walkowiak [29], Ellenberg-Venkatesh [11,Remark 2.3], Binyamini and Novikov [2,Theorem 6], and others.…”

“…Heath-Brown [17] introduces a form of this conjecture with uniformity in X for fixed d and n, and he develops a new variant of the determinant method using p-adic approximation instead of smooth parameterizations as in [3], [2], [23], [10]. In [26], Salberger proves this uniform version of the dimension growth conjecture for d ≥ 4.…”