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

Betti number;Homotopy type;Quadratic surfaces;Quadratic polynomials

Kettner, Michael ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Betti number;    Homotopy type;    Quadratic surfaces;    Quadratic polynomials;   

Others  :  https://smartech.gatech.edu/bitstream/1853/19704/1/kettner_michael_200712_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

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.