Topological lower bounds for arithmetic networks

Abstract

We prove that the depth of any arithmetic network for deciding membership in a semialgebraic set ${\Sigma \subset \mathbb{R}^{n}}$ is bounded from below by

$$c_1 \sqrt{ \frac{\log ({\rm b}(\Sigma))}{n}} -c_2 \log n,$$

where ${{\rm b}(\Sigma)}$ is the sum of the Betti numbers of ${\Sigma}$ with respect to “ordinary” (singular) homology, and c ₁, c ₂ 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 ${\rho: \mathbb{R}^{n} \to \mathbb{R}^{n-r}}$ is the projection map, for some ${r=0, \ldots, n}$, then the depth of any arithmetic network deciding membership in ${\Sigma}$ is bounded by

$$\frac{c_1\sqrt{\log ({\rm b}(\rho(\Sigma)))}}{n} - c_2 \log n$$

for some positive constants c ₁, c ₂.