Systems of Quadratic Inequalities

TL;DR

This paper introduces a spectral sequence method for efficiently computing Betti numbers and homology images of semi-algebraic sets defined by quadratic inequalities in real projective space.

Contribution

It provides a novel spectral sequence approach with an explicit formula for the differential d_2, improving computational techniques for quadratic inequality systems.

Findings

Efficient computation of Betti numbers for quadratic inequality sets

Explicit formula for the differential d_2 in the spectral sequence

Applicable to homology homomorphisms induced by inclusions

Abstract

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.