AbstractWe prove that the depth of any arithmetic network for deciding membership in a semialgebraic set $${\Sigma \subset \mathbb{R}^{n}}$$Σ⊂Rn is bounded from below by $$c_1 \sqrt{ \frac{\log ({\rm b}(\Sigma))}{n}} -c_2 \log n,$$c1log(b(Σ))n-c2logn, where $${{\rm b}(\Sigma)}$$b(Σ) is the sum of the Betti numbers of $${\Sigma}$$Σ with respect to “ordinary” (singular) homology, and c1, c2 are some (absolute) positive constants. This result complements the similar lower bound in Montaña et al. (Appl Algebra Engrg Comm Comput 7:41–51, 1996) for locally closed semialgebraic sets in terms of the sum of Borel–Moore Betti numbers.We also prove that if $${\rho: \mathbb{R}^{n} \to \mathbb{R}^{n-r}}$$ρ:Rn→Rn-r is the projection map, for some $${r=0, \ldots, n}$$r=0,…,n, then the depth of any arithmetic network deciding membership in $${\Sigma}$$Σ is bounded by $$\frac{c_1\sqrt{\log ({\rm b}(\rho(\Sigma)))}}{n} - c_2 \log n$$c1log(b(ρ(Σ)))n-c2lognfor some positive constants c1, c2.

中文翻译：

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算术网络的拓扑下界

摘要我们证明了用于决定半代数集合 $${\Sigma \subset \mathbb{R}^{n}}$$Σ⊂Rn 的任何算术网络的深度都由 $$c_1 \sqrt{ \ frac{\log ({\rm b}(\Sigma))}{n}} -c_2 \log n,$$c1log(b(Σ))n-c2logn，其中 $${{\rm b}(\ Sigma)}$$b(Σ) 是 $${\Sigma}$$Σ 关于“普通”（奇异）同源性的 Betti 数之和，c1、c2 是一些（绝对）正常数。这一结果补充了 Montaña 等人的类似下限。(Appl Algebra Engrg Comm Comput 7:41–51, 1996) 根据 Borel–Moore Betti 数之和的局部封闭半代数集。我们还证明如果 $${\rho: \mathbb{R}^{n } \to \mathbb{R}^{nr}}$$ρ:Rn→Rn-r 是投影图，对于某些 $${r=0, \ldots, n}$$r=0,...,n ,