# An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic Set

## TL;DR

This paper introduces an adaptive grid algorithm on the unit sphere to improve the efficiency of computing homology groups of semialgebraic sets, advancing practical methods in real algebraic geometry.

## Contribution

It presents a novel adaptive grid algorithm that enhances the existing weak exponential time algorithm for homology computation of semialgebraic sets.

## Key findings

- Improved algorithm reduces computational complexity.
- Demonstrates practical efficiency on example sets.
- Advances the applicability of homology computation methods.

## Abstract

Looking for an efficient algorithm for the computation of the homology groups of an algebraic set or even a semi-algebraic set is an important problem in the effective real algebraic geometry. Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez wrote a paper [1], which made a step forward by giving an algorithm of weak exponential time. However, the algorithm is not not practical. In my thesis, I will introduce my improvement of this algorithm using an adaptive grid algorithm on the unit sphere.