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Complex cellular structures. (English) Zbl 1448.14056
In this outstanding paper, the notion of a complex cell is introduced. Complex cells are complexifications of the cells used in real tame geometry (semialgebraic geometry or more general o-minimal geometry).
Here are the definitions. For $$r\in \mathbb{C}$$ with $$|r| > 0$$ let $$D(r) := \{|z|<|r|\}$$ and $$D_\circ(r) := \{0<|z|<|r|\}$$. For $$r_1, r_2\in \mathbb{C}$$ with $$|r_2| > |r_1| > 0$$ let $$A(r_1, r_2) :=\{|r_1|<|z|<|r_2|\}$$. Moreover, let $$*:=\{0\}$$. For $$0 < \delta < 1$$ the $$\delta$$-extensions are defined by $$D^\delta(r) := D(\delta^{-1}r),D^\delta_\circ (r):=D_\circ(\delta^{-1} r), A^\delta(r_1, r_2) := A(\delta r_1, \delta^{-1}r_2)$$ and $$*^\delta:=*$$. These correspond to the Euclidean geometry of the complex plane. In many cases the following extensions are more suitable which reflect the hyperbolic geometry of the domains. For $$0 < \rho < \infty$$ the $$\{\rho\}$$-extension $$\mathcal{F}^{\{\rho\}}$$ of $$\mathcal{F}$$ is given by $$\mathcal{F}^\delta$$ where $$\delta$$ satisfies $$\rho=2\pi\delta/(1-\delta)$$ if $$\mathcal{F}$$ is of type $$D$$ and $$\rho=\pi^2/(2|\log \delta|)$$ if $$\mathcal{F}$$ is of type $$D_\circ$$ or $$A$$. The motivation for this notation comes from the following.
Fact. Let $$\mathcal{F}$$ be a domain of type $$A,D,D_\circ$$, and let $$S$$ be a component of the boundary of $$\mathcal{F}$$ in $$\mathcal{F}^{\{\rho\}}$$. Then the length of $$S$$ in $$\mathcal{F}^{\{\rho\}}$$ is at most $$\rho$$.
For the definition of a complex cell the following notation is used. Let $$\mathcal{X},\mathcal{Y}$$ be sets and $$\mathcal{F}:\mathcal{X}\to 2^{\mathcal{Y}}$$ be a map taking points of $$\mathcal{X}$$ to subsets of $$\mathcal{Y}$$. Then $$\mathcal{X}\odot\mathcal{F}:=\{(x,y)\colon x\in\mathcal{X}, y\in \mathcal{F}(x)\}$$. Here $$\mathcal{X}$$ will be a subset of $$\mathbb{C}^n$$ and $$\mathcal{Y}$$ will be $$\mathbb{C}$$. If $$r:\mathcal{X}\to \mathbb{C}\setminus\{0\}$$ is a map then $$D(r)$$ stands for the map which assigns to $$x\in\mathcal{X}$$ the disc $$D(r(x))$$, and similarly for $$D_\circ$$ and $$A$$. If $$U$$ is a complex manifold the space of holomorphic functions on $$U$$ is denoted by $$\mathcal{O}(U)$$. By $$\mathcal{O}_b(U)$$ the subspace of bounded holomorphic functions on $$U$$ is denoted. The symbol $$z_{1..\ell}$$ stands for the tuple $$(z_1,\ldots,z_l)$$ of variables. Now the definition of a complex cell $$\mathcal{C}$$ of length $$\ell\in\mathbb{N}_0$$ is as follows.
Definition. A complex cell $$\mathcal{C}$$ of length zero is the point $$\mathbb{C}^0$$. A complex cell of length $$\ell + 1$$ has the form $$\mathcal{C}_{1..\ell}\odot \mathcal{F}$$ where the base $$\mathcal{C}_{1..\ell}$$ is a complex cell of length $$\ell$$ and the fiber $$\mathcal{F}$$ is one of $$*,D(r),D_\circ(r),A(r_1, r_2)$$ where $$r\in \mathcal{O}_b(\mathcal{C}_{1..\ell})$$ satisfies $$|r(z_{1..\ell})|>0$$ for $$z_{1..\ell}\in\mathcal{C}_{1..\ell}$$; and $$r_1,r_2\in \mathcal{O}_b(\mathcal{C}_{1..\ell})$$ satisfy $$|r_2(z_{1..\ell}|>|r_1(z_{1..\ell})|>0$$ for $$z_{1..\ell}\in \mathcal{C}_{1..\ell}$$.
The notion of a $$\delta$$-extension of a complex cell of length $$\ell$$ is defined as follows.
Definition. The cell of length zero is its own $$\delta$$-extension. A cell $$\mathcal{C}$$ of length $$\ell + 1$$ admits a $$\delta$$-extension $$\mathcal{C}^\delta:=C_{1..\ell}^\delta\odot \mathcal{F}^\delta$$ if $$\mathcal{C}_{1..\ell}$$ admits a $$\delta$$-extension, and if the function $$r$$ (resp. $$r_1,r_2$$) involved in $$\mathcal{F}$$ admits holomorphic continuation to $$\mathcal{C}^\delta_{1..\ell}$$ and satisfies $$|r(z_{1..\ell})| > 0$$ (resp. $$|r_2(z_{1..\ell})| > |r_1(z_{1..\ell})| > 0$$) in this larger domain.
The $$\{\rho\}$$-extension $$\mathcal{C}^{\{\rho\}}$$ is defined in an analogous manner. The category of complex cells is equipped with cellular maps.
Definition. Let $$\mathcal{C},\hat{\mathcal{C}}$$ be cells of length $$\ell$$. A holomorphic map $$f:\mathcal{C}\to\hat{\mathcal{C}}$$ is cellular if it takes the form $$w_j = \phi_j(z_{1..j})$$, where $$\phi_j\in\mathcal{O}_b(\mathcal{C}_{1..j})$$ is a monic polynomial of positive degree in $$z_j$$ for $$j = 1,\ldots,\ell$$.
In the above situation the map $$f$$ is prepared if, for $$j = 1,\ldots,\ell$$, it is $$\phi_j(z_{1..j})=z_j^{q_j} + \tilde{\phi}_j(z_{1..j-1})$$ for some $$q_j\in \mathbb{N}$$ and holomorphic $$\tilde{\phi}_j$$. The main result of the paper is the Cellular Parametrization Theorem, CPT. We formulate it here in several statements.
Qualitative CPT. Let $$\rho,\sigma\in (0,\infty)$$. Let $$\mathcal{C}$$ be a complex cell which admits a $$\{\rho\}$$-extension. Let $$F_1,\ldots,F_M \in \mathcal{O}_b(\mathcal{C}^{\{\rho\}})$$. Then there is a finite list $$\mathcal{C}_1,\ldots,\mathcal{C}_N$$ of complex cells which admit $$\{\sigma\}$$-extensions and, for each $$j = 1,\ldots,N$$, a cellular map $$f_j:\mathcal{C}^{\{\sigma\}}\to\mathcal{C}^{\{\rho\}}$$ such that the following properties hold:
(1) $$\mathcal{C}\subset \bigcup_j f_j(\mathcal{C}_j)$$;
(2) $$f_j$$ is prepared for every $$j$$;
(3) $$F_k\circ f_j$$ vanishes either identically or nowhere for every $$k,j$$.
The quantitative versions of the Cellular Parametrization Theorem give upper bound for the above size $$N$$ of the cover. In the first version definable means definable in the o-minimal structure $$\mathbb{R}_{\mathrm{an}}$$ of restricted analytic functions (or equivalently, the structure generated by the bounded subanalytic sets).
Quantitative CPT I. If $$C^{\{\rho\}}$$ vary in a definable family $$\Lambda$$ then there is for every parameter $$\lambda\in\Lambda$$ a polynomial $$P_\lambda(X, Y )$$ with positive coefficients such that there is a cover of size $$N_\lambda\leq P_\lambda(\rho,1/\sigma)$$. Moreover, the covering functions can be chosen from a single definable family.
Quantitative CPT II. There are polynomials $$P,Q$$ with positive coefficients depending on the length of the cell with the following properties. If $$\mathcal{C}^{\{\rho\}}, F_1,\ldots,F_M$$ are algebraic of complexity $$\beta$$ there is a cover of size bounded by $$P(\beta,N,\rho,1/\sigma)$$ and complexity bounded by $$Q(M,\beta)$$.
Using the power of complex analysis as for instance the Cauchy estimates to control the derivatives the following central result is obtained, providing a strong improvement on the bounds in the Yomdin-Gromov Algebraic Lemma on the parametrization of semialgebraic sets (see [M. Gromov: Astérisque 145–146, 225–240, Exp. No. 663 (1987; Zbl 0611.58041]).
Refined Algebraic Lemma. Let $$X = \{X_p\subset [0,1]^n\}$$ be a semialgebraic family, with $$\dim X_p\leq\mu$$. Set $$B = (0, 1)^\mu$$. There exist constants $$C = C(X)$$ and $$\varepsilon=\varepsilon(X)$$ such that for any $$p$$, there exist maps $$\phi_1,\ldots,\phi_C:B\to X_p$$ whose images cover $$X_p$$. Moreover, there are polynomials $$P,Q$$ depending on $$n$$ such that $$C\leq P(\beta)$$ and $$\varepsilon^{-1}\leq Q(\beta)$$ where $$\beta$$ is the complexity of the semialgebraic family $$X$$.
The Refined Algebraic Lemma settles a conjecture of Yomdin on the topological entropy of analytic maps, using the work of D. Burguet et al. [Proc. Lond. Math. Soc. (3) 111, No. 2, 381–419 (2015; Zbl 1352.37015)]. Note that there is an appendix by Yomdin to the article giving a direct proof of his conjecture starting from the Refined Algebraic Lemma.
The Refined Algebraic Lemma and its also established subanalytic version lead also to new results related to the Pila-Wilkie theorem (see [J. Pila and A. J. Wilkie, Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)]) and imply as this celebrated result applications to unlikely intersections in diophantine geometry (see for example [J. Pila, Ann. Math. (2) 173, No. 3, 1779–1840 (2011; Zbl 1243.14022)]).

##### MSC:
 14P10 Semialgebraic sets and related spaces 37B40 Topological entropy 03C64 Model theory of ordered structures; o-minimality 30C99 Geometric function theory
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