1 Introduction to the Method.- 2 Importance of QE and CAD Algorithms.- 3 Alternative Approaches.- 4 Practical Issues.- Acknowledgments.- Quantifier Elimination by Cylindrical Algebraic Decomposition - Twenty Years of Progress.- 1 Introduction.- 2 Original Method.- 3 Adjacency and Clustering.- 4 Improved Projection.- 5 Partial CADs.- 6 Interactive Implementation.- 7 Solution Formula Construction.- 8 Equational Constraints.- 9 Subalgorithms.- 10 Future Improvements.- A Decision Method for Elementary Algebra and Geometry.- 1 Introduction.- 2 The System of Elementary Algebra.- 3 Decision Method for Elementary Algebra.- 4 Extensions to Related Systems.- 5 Notes.- 6 Supplementary Notes.- Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Algebraic Foundations.- 3 The Main Algorithm.- 4 Algorithm Analysis.- 5 Observations.- Super-Exponential Complexity of Presburger Arithmetic.- 1 Introduction and Main Theorems.- 2 Algorithms.- 3 Method for Complexity Proofs.- 4 Proof of Theorem 3 (Real Addition).- 5 Proof of Theorem 4 (Lengths of Proofs for Real Addition).- 6 Proof of Theorems 1 and 2 (Presburger Arithmetic).- 7 Other Results.- Cylindrical Algebraic Decomposition I: The Basic Algorithm.- 1 Introduction.- 2 Definition of Cylindrical Algebraic Decomposition.- 3 The Cylindrical Algebraic Decomposition Algorithm: Projection Phase.- 4 The Cylindrical Algebraic Decomposition Algorithm: Base Phase.- 5 The Cylindrical Algebraic Decomposition Algorithm: Extension Phase.- 6 An Example.- Cylindrical Algebraic Decomposition II: An Adjacency Algorithm for the Plane.- 1 Introduction.- 2 Adjacencies in Proper Cylindrical Algebraic Decompositions.- 3 Determination of Section-Section Adjacencies.- 4 Construction of Proper Cylindrical Algebraic Decompositions.- 5 An Example.- An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Idea.- 3 Analysis.- 4 Empirical Results.- Partial Cylindrical Algebraic Decomposition for Quantifier Elimination.- 1 Introduction.- 2 Main Idea.- 3 Partial CAD Construction Algorithm.- 4 Strategy for Cell Choice.- 5 Illustration..- 6 Empirical Results.- 7 Conclusion.- Simple Solution Formula Construction in Cylindrical Algebraic Decomposition Based Quantifier Elimination.- 1 Introduction.- 2 Problem Statement.- 3 (Complex) Solution Formula Construction.- 4 Simplification of Solution Formulas.- 5 Experiments.- Recent Progress on the Complexity of the Decision Problem for the Reals.- 1 Some Terminology.- 2 Some Complexity Highlights.- 3 Discussion of Ideas Behind the Algorithms.- An Improved Projection Operation for Cylindrical Algebraic Decomposition.- 1 Introduction..- 2 Background Material..- 3 Statements of Theorems about Improved Projection Map.- 4 Proof of Theorem 3 (and Lemmas).- 5 Proof of Theorem 4 (and Lemmas).- 6 CAD Construction Using Improved Projection.- 7 Examples.- 8 Appendix.- Algorithms for Polynomial Real Root Isolation.- 1 Introduction.- 2 Preliminary Mathematics.- 3 Algorithms.- 4 Computing Time Analysis.- 5 Empirical Computing Times.- Sturm- Twenty Years of Progress.- 1 Introduction.- 2 Original Method.- 3 Adjacency and Clustering.- 4 Improved Projection.- 5 Partial CADs.- 6 Interactive Implementation.- 7 Solution Formula Construction.- 8 Equational Constraints.- 9 Subalgorithms.- 10 Future Improvements.- A Decision Method for Elementary Algebra and Geometry.- 1 Introduction.- 2 The System of Elementary Algebra.- 3 Decision Method for Elementary Algebra.- 4 Extensions to Related Systems.- 5 Notes.- 6 Supplementary Notes.- Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Algebraic Foundations.- 3 The Main Algorithm.- 4 Algorithm Analysis.- 5 Observations.- Super-Exponential Complexity of Presburger Arithmetic.- 1 Introduction and Main Theorems.- 2 Algorithms.- 3 Method for Complexity Proofs.- 4 Proof of Theorem 3 (Real Addition).- 5 Proof of Theorem 4 (Lengths of Proofs for Real Addition).- 6 Proof of Theorems 1 and 2 (Presburger Arithmetic).- 7 Other Results.- Cylindrical Algebraic Decomposition I: The Basic Algorithm.- 1 Introduction.- 2 Definition of Cylindrical Algebraic Decomposition.- 3 The Cylindrical Algebraic Decomposition Algorithm: Projection Phase.- 4 The Cylindrical Algebraic Decomposition Algorithm: Base Phase.- 5 The Cylindrical Algebraic Decomposition Algorithm: Extension Phase.- 6 An Example.- Cylindrical Algebraic Decomposition II: An Adjacency Algorithm for the Plane.- 1 Introduction.- 2 Adjacencies in Proper Cylindrical Algebraic Decompositions.- 3 Determination of Section-Section Adjacencies.- 4 Construction of Proper Cylindrical Algebraic Decompositions.- 5 An Example.- An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition.- 1 Introduction.- 2 Idea.- 3 Analysis.- 4 Empirical Results.- Partial Cylindrical Algebraic Decomposition for Quantifier Elimination.- 1 Introduction.- 2 Main Idea.- 3 Partial CAD Construction Algorithm.- 4 Strategy for Cell Choice.- 5 Illustration..- 6 Empirical Results.- 7 Conclusion.- Simple Solution Formula Construction in Cylindrical Algebraic Decomposition Based Quantifier Elimination.- 1 Introduction.- 2 Problem Statement.- 3 (Complex) Solution Formula Construction.- 4 Simplification of Solution Formulas.- 5 Experiments.- Recent Progress on the Complexity of the Decision Problem for the Reals.- 1 Some Terminology.- 2 Some Complexity Highlights.- 3 Discussion of Ideas Behind the Algorithms.- An Improved Projection Operation for Cylindrical Algebraic Decomposition.- 1 Introduction..- 2 Background Material..- 3 Statements of Theorems about Improved Projection Map.- 4 Proof of Theorem 3 (and Lemmas).- 5 Proof of Theorem 4 (and Lemmas).- 6 CAD Construction Using Improved Projection.- 7 Examples.- 8 Appendix.- Algorithms for Polynomial Real Root Isolation.- 1 Introduction.- 2 Preliminary Mathematics.- 3 Algorithms.- 4 Computing Time Analysis.- 5 Empirical Computing Times.- Sturm-Habicht Sequences, Determinants and Real Roots of Univariate Polynomials.- 1 Introduction.- 2 Algebraic Properties of Sturm-Habicht Sequences.- 3 Sturm-Habicht Sequences and Real Roots of Polynomial.- 4 Sturm-Habicht Sequences and Hankel Forms.- 5 Applications and Examples.- Characterizations of the Macaulay Matrix and Their Algorithmic Impact.- 1 Introduction.- 2 Notation.- 3 Definitions of the Macaulay Matrix.- 4 Extraneous Factor and First Properties of the Macaulay Determinant.- 5 Characterization of the Macaulay Matrix.- 6 Characterization of the Macaulay Matrix, if It Is Used to Calculate the u-Resultant.- 7 Two Sorts of Homogenization.- 8 Characterization of the Matrix of the Extraneous Factor.- 9 Conclusion.- Computation of Variant Resultants.- 1 Introduction.- 2 Problem Statement.- 3 Review of Determinant Based Method.- 4 Quotient Based Method.- 5 Modular Methods..- 6 Theoretical Computing Time Analysis.- 7 Experiments.- A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials.- 1 Introduction.- 2 Proof of the Theorem.- Local Theories and Cylindrical Decomposition.- 1 Introduction.- 2 Infinitesimal Sectors at the Origin.- 3 Neighborhoods of Infinity.- 4 Exponential Polynomials in Two Variables.- A Combinatorial Algorithm Solving Some Quantifier Elimination Problems.- 1 Introduction.- 2 Sturm-Habicht Sequence.- 3 The Algorithms.- 4 Conclusions.- A New Approach to Quantifier Elimination for Real Algebra.- 1 Introduction.- 2 The Quantifier Elimination Problem for the Elementary Theory of the Reals.- 3 Counting Real Zeros Using Quadratic Forms.- 4 Comprehensive Grobner Bases.- 5 Steps of the Quantifier Elimination Method.- 6 Examples.- References.

Algorithms in Real Algebraic Geometry and Applications to Computational Geometry.

Quantifier elimination and cylindrical algebraic decomposition

background

Algorithmic algebra

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in Chapter 1 of this Handbook, and are extended in this chapter to higher dimensions.

Arrangements and Their Applications

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.

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

Let $F$ and $G$ be two collections of a total of $n$ bivariate algebraic functions of constant maximum degree. The minimization diagrams of $F$, $G$ are the planar maps obtained by the $xy$-projections of the lower envelopes of $F$, $G$, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of $F$ and of $G$ is $O(n^{2+\varepsilon})$, for any $\varepsilon<0$. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopesein three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply-shaped convex sets in three dimensions.

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Algorithms in Real Algebraic Geometry and Applications to Computational Geometry.

Citations

Quantifier elimination and cylindrical algebraic decomposition

Algorithmic algebra

Arrangements and Their Applications

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

The Overlay of Lower Envelopes and its Applications

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