# Computing the Homology of Semialgebraic Sets. II: General formulas

## TL;DR

This paper presents a numerical algorithm for computing the homology of semialgebraic sets with complexity that is single exponential outside a negligible subset, improving over previous doubly exponential methods.

## Contribution

The authors extend their previous work to arbitrary semialgebraic sets, providing a more efficient algorithm with weak exponential time complexity.

## Key findings

- Algorithm computes Betti numbers and torsion coefficients.
- Works in weak exponential time outside a small measure subset.
- Improves complexity from doubly exponential to single exponential.

## Abstract

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the previous work of the authors in arXiv:1807.06435 to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity.