Topology of quadratic maps and hessians of smooth maps

In this paper we study sets of real solutions of systems of quadratic equations and inequalities. The results are used for the local study of more general systems of smooth equations and inequalities.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 26, pp. 85–124, 1988.