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 Betti numbers as well as on the number of different stable homotopy types of certain fibers 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. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.
Deep Dive into Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial.
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 Betti numbers as well as on the number of different stable homotopy types of certain fibers 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. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm im
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 the following new bounds on the topological complexity of semialgebraic 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.
(1) Let S ⊂ R k be defined by P 1 ≥ 0, . . . , P m ≥ 0 with P i ∈ R[X 1 , . . . , X k ], m < k, and deg(P i ) ≤ 2, for 1 ≤ i ≤ m. We prove that b i (S)
(2) Let P = {P 1 , . . . , P m } ⊂ R[Y 1 , . . . , Y ℓ , X 1 , . . . , X k ], with deg Y (P i ) ≤ 2, deg X (P i ) ≤ d, 1 ≤ i ≤ m. Let S ⊂ R ℓ+k be a semi-algebaic set, defined by a Boolean formula without negations, whose atoms are of the form, P ≥ 0, P ≤ 0, P ∈ P.
Let π : R ℓ+k → R k be the projection on the last k co-ordinates. We prove that the number of stable homotopy types amongst the fibers π -1 (x) ∩ S is bounded by (2 m ℓkd) O(mk) . We conclude the thesis with presenting two new algorithms along with their implementations. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in R k which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in R 3 using a semi-numerical approach.
The writing of this thesis has been one of the most significant academic challenges I have had to face. Without the support, patience and guidance of the following people and institutes, this study would not have been completed. It is to them that I owe my deepest gratitude.
In the first place I would like to record my gratitude to Saugata Basu for his supervision. His wisdom, knowledge and commitment to the highest standards inspired and motivated me. Moreover, he always gave me a lot of freedom and made it possible to visit many interesting places in order to learn from many outstanding researchers. I am indebted to him more than he knows.
I gratefully acknowledge Laureano González-Vega for his advice and supervision of my research during my two year visit to the Universidad de Cantabria in Santander, Spain, which was supported by the European RTNetwork Real Algebraic and Analytic Geometry (Contract No. HPRN-CT-2001-00271). He introduced me to another very exciting area of Real Algebraic Geometry. In additon, he serves on my committee.
Many thanks also go to John Etnyre, Mohammad Ghomi and Victoria Powers for serving on my committee.
It is a pleasure to pay tribute also to the entire staff, especially to Genola Turner, and the professors, especially to Alfred Andrew, Eric Carlen, Luca Dieci, Wilfrid Gangbo and William Green, of the School of Mathematics. Furthermore, I would like to thank Tomás Recio Muñiz and Fernando Etayo Gordejuela from the Universidad de Cantabria. They all have provided an environment that is both supportive and intellectually stimulating.
I am very grateful to the European RTNetwork Real Algebraic and Analytic Geometry (Contract No. HPRN-CT-2001-00271), the Institut Henri Poincaré in Paris, France, the Institute for Mathematics and its Applications in Minneapolis, MN, and the Mathematical Sciences Research Institute in Berkeley, CA, for giving me the opportunity to visit several workshops and meeting many outstanding researchers. Moreover, I would like to thank Chris Brown for his steady help with QEPCAD B, Michel Coste for simplifying the proof of Proposition 2.24, Ioana Necula for providing her source code of the TOP-algorithm and Nicola Wolpert for very useful discussions and comments. In classical algebraic geometry, the main objects of interest are complex algebraic sets, i.e. the zero set of a finite family of polynomials over the field C of complex numbers, meaning the set of all points that simultaneously satisfy one or more polynomial equations. But in many applications in computer-aided geometric design, computational geometry, robotics or computer graphics one is interested in the solutions over the field R of real numbers. Moreover, they also deal with the real solutions of finite systems of inequalities which are the main objects of real algebraic geometry. Unfortunately, real algebraic sets have a very different behavior than their complex counterparts. For example, an irreducible algebraic subset of C k having complex dimension n, considered as an algebraic subset of R 2k is connected, not bounded (unless it is a point) and has local real dimension 2n at every point (see, f
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