Computing the Homology of Real Projective Sets
This work provides the first numerically stable algorithm for homology computation of real projective sets, addressing a fundamental problem in computational topology for practitioners.
The paper presents a numerical algorithm for computing homology (Betti numbers and torsion coefficients) of real projective varieties, with cost singly exponential in the number of variables and polynomial in condition and degrees. It also shows that outside a measure-zero exceptional set, the algorithm runs in exponential time.
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.