复杂的细胞结构
Complex Cellular Structures
摘要：
我们介绍了一个复杂的细胞的概念，一个复杂的细胞/圆柱体中使用的真正驯服几何。对于$δ\in(0,1)$和复胞元$\mathcal{C}$，我们定义它的全纯扩张$\mathcal{C}\subset\mathcal{C}^δ$，它也是复胞元。$\mathcal{C}^δ$中的$\mathcal{C}$的双曲几何为复杂胞元类提供了在实际情况中不存在的丰富的几何函数理论。我们用它来证明一个复杂的模拟真正驯服几何的细胞分解定理。在代数的情况下，我们表明，这种分解的复杂性多项式依赖于所涉及的方程的程度。利用这一理论，我们完善的Yomdin-Gromov代数引理的$C^r$ -光滑参数化的半代数集：我们表明，$C^r$图表的数量可以采取多项式的光滑顺序$r$和在复杂的集合。代数引理最初是在Yomdin和Gromov的工作中发明的，用于估计$C^\infty$映射的拓扑熵。对于解析映射，我们的改进版本，结合工作的Burguet，廖和杨，建立了一个最佳的细化这些估计的形式紧边界的尾部熵和体积增长。这解决了一个猜想的Yomdin谁证明了同样的结果，在第二维在1991年。Yomdin在附录中给出了使用精炼代数引理的这些估计的自包含证明。代数引理最近被用于研究代数簇和超越簇上的有理点。我们在这两个方向上使用复杂细胞理论。在代数的背景下，我们改进了Heath-Brown关于代数簇中插值有理点的一个结果。在超越的情况下，我们证明了插值结果（无限制）对数图像的次解析集。
英文摘要：
We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $δ\in(0,1)$ and a complex cell $\mathcal{C}$ we define its holomorphic extension $\mathcal{C}\subset\mathcal{C}^δ$, which is again a complex cell. The hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^δ$ 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 $C^r$-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. 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.
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