Topological lower bounds for arithmetic networks

TL;DR

This paper establishes topological lower bounds on the complexity of deciding membership in semialgebraic sets using Betti numbers, linking algebraic topology with computational complexity in arithmetic networks.

Contribution

It introduces new lower bounds based on Betti numbers for arithmetic network decision problems, extending previous work to include ordinary homology and projections.

Findings

Lower bounds are expressed in terms of the sum of Betti numbers.

Results apply to both local and projected semialgebraic sets.

The bounds complement existing results by Montana, Morais, and Pardo.

Abstract

We prove a complexity lower bound on deciding membership in a semialgebraic set for arithmetic networks in terms of the sum of Betti numbers with respect to "ordinary" (singular) homology. This result complements a similar lower bound by Montana, Morais and Pardo for locally close semialgebraic sets in terms of the sum of Borel-Moore Betti numbers. We also prove a lower bound in terms of the sum of Betti numbers of the projection of a semialgebraic set to a coordinate subspace.