Saugata Basu

Bio: Saugata Basu is an academic researcher from Purdue University. The author has contributed to research in topics: Betti number & Algebraic number. The author has an hindex of 27, co-authored 155 publications receiving 4077 citations. Previous affiliations of Saugata Basu include Courant Institute of Mathematical Sciences & Georgia Institute of Technology.

Algorithms in real algebraic geometry

Abstract: Algebraically Closed Fields- Real Closed Fields- Semi-Algebraic Sets- Algebra- Decomposition of Semi-Algebraic Sets- Elements of Topology- Quantitative Semi-algebraic Geometry- Complexity of Basic Algorithms- Cauchy Index and Applications- Real Roots- Cylindrical Decomposition Algorithm- Polynomial System Solving- Existential Theory of the Reals- Quantifier Elimination- Computing Roadmaps and Connected Components of Algebraic Sets- Computing Roadmaps and Connected Components of Semi-algebraic Sets

On the combinatorial and algebraic complexity of quantifier elimination

Abstract: In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

Abstract: In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Given a closed semi-algebraic set S $\subset$ R k defined as the intersection of a real variety, Q=0, deg(Q)≤d, whose real dimension is k', with a set defined by a quantifier-free Boolean formula with no negations with atoms of the form P i =0, P i ≥ 0, P i ≤ 0, deg(P i ) ≤ d, 1≤ i≤ s, we prove that the sum of the Betti numbers of S is bounded by s k' (O(d)) k . This result generalizes the Oleinik—Petrovsky—Thom—Milnor bound in two directions. Firstly, our bound applies to arbitrary unions of basic closed semi-algebraic sets, not just for basic semi-algebraic sets. Secondly, the combinatorial part (the part depending on s ) in our bound, depends on the dimension of the variety rather than that of the ambient space. It also generalizes the result in [4] where a similar bound is proven for the number of connected components. We also prove that the sum of the Betti numbers of S is bounded by s k' 2 O(k2 m4) in case the total number of monomials occurring in the polynomials in $ {\cal P} \cup \{Q\}$ is m. Using the tools developed for the above results, as well as some additional techniques, we give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets.

On the combinatorial and algebraic complexity of quantifier elimination

Abstract: In this paper we give a new algorithm for performing quantifier elimination from first order formulae over real closed fields. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of our algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated, making possible our improved complexity bound. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that we output, are independent of the number of input polynomials. As special cases of this algorithm, we obtain new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields. Using the theory developed in this paper, we also give an improved bound on the radius of a ball centered at the origin, which is guaranteed to intersect every connected component of the sign partition induced by a family of polynomials. We also use our methods to obtain algorithms for solving certain decision problems in real and complex geometry which improves the complexity of the currently known algorithms for these problems. >

Principles of Robot Motion: Theory, Algorithms, and Implementations

Abstract: A text that makes the mathematical underpinnings of robot motion accessible and relates low-level details of implementation to high-level algorithmic concepts. Robot motion planning has become a major focus of robotics. Research findings can be applied not only to robotics but to planning routes on circuit boards, directing digital actors in computer graphics, robot-assisted surgery and medicine, and in novel areas such as drug design and protein folding. This text reflects the great advances that have taken place in the last ten years, including sensor-based planning, probabalistic planning, localization and mapping, and motion planning for dynamic and nonholonomic systems. Its presentation makes the mathematical underpinnings of robot motion accessible to students of computer science and engineering, rleating low-level implementation details to high-level algorithmic concepts.

Semidefinite programming relaxations for semialgebraic problems

Abstract: A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.

Complexity and Real Computation

Abstract: 1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Lectures on Discrete Geometry

Abstract: From the Publisher: Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Computing Persistent Homology

Abstract: We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.