Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

TL;DR

This paper introduces a singly exponential algorithm for computing the first Betti number of semi-algebraic sets and describes connected components efficiently, advancing computational topology in real algebraic geometry.

Contribution

It presents the first singly exponential algorithm for the first Betti number and connected components of semi-algebraic sets, improving computational methods in the field.

Findings

Singly exponential algorithm for the first Betti number.

Efficient semi-algebraic descriptions of connected components.

Improved computational complexity over previous methods.

Abstract

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results.