Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomials

Abstract

In this thesis, we consider semi-algebraic sets over a real closed field R defined by quadratic polynomials. Semi-algebraic sets of R^k are defined as the smallest family of sets in R^k that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the topological complexity of semi-algebraic sets over a real closed field R defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations.