Computing the Homology of Real Projective Sets

TL;DR

This paper introduces a numerically stable algorithm for computing the homology of real projective varieties, with complexity depending on input condition and size, and analyzes its performance and limitations.

Contribution

It presents a new numerical algorithm for homology computation of real projective sets, analyzing its stability, complexity, and performance bounds.

Findings

Algorithm is numerically stable and efficient for certain inputs.

Complexity depends exponentially on the ambient space dimension.

Outside a small measure set, the algorithm's runtime is exponential.

Abstract

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.