 
Summary: arXiv:1012.5731v2[math.AG]30Dec2010
SYSTEMS OF QUADRATIC INEQUALITIES
A. AGRACHEV AND A. LERARIO
Abstract. We present a spectral sequence which efficiently computes Betti
numbers of a closed semialgebraic subset of RPn
defined by a system of qua
dratic inequalities and the image of the homology homomorphism induced by
the inclusion of this subset in RPn. We do not restrict ourselves to the term E2
of the spectral sequence and give a simple explicit formula for the differential
d2.
1. Introduction
In this paper we study closed semialgebraic subsets of RPn
presented as the
sets of solutions of systems of homogeneous quadratic inequalities. Systems are
arbitrary: no regularity condition is required and systems of equations are included
as special cases. Needless to say, standard Veronese map reduces any system of
homogeneous polynomial inequalities to a system of quadratic ones (but the number
of inequalities in the system increases). The nonhomogeneous affine case will be
the subject of another publication.
To study a system of quadratic inequalities we focus on the dual object. Namely,
