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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)]).

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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