关于可定义映射纤维的同伦型数
On the number of homotopy types of fibres of a definable map
摘要：
在本文中，我们证明了一个单一的指数上界的数目可能同伦类型的Pfavian映射的纤维，在其图形的格式。特别是，我们表明，如果一个半代数集$S \subset {\R}^{m+n}$，其中$\R$是一个真正的封闭领域，是由一个布尔公式定义的$s$多项式的次数小于$d$，和$π: {\R}^{m+n} \to {\R}^n$是在一个子空间上的投影，那么不同同伦类型的纤维的$π$的数目不超过$s^{2(m+1)n}(2^m nd)^{O(nm)}$。作为我们主要结果的应用，我们证明了由fewnomials定义的半代数集的同伦类型数的单指数界，以及由具有有界可加复杂性的多项式定义的半代数集的同伦类型数的单指数界。我们还证明了单指数上界的半径保证局部压缩的半代数集定义的整系数多项式。
英文摘要：
In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \subset {\R}^{m+n}$, where $\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $π: {\R}^{m+n} \to {\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $π$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials with bounded additive complexity. We also prove single exponential upper bounds on the radii of balls guaranteeing local contractibility for semi-algebraic sets defined by polynomials with integer coefficients.
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