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23
On the Combinatorial and Algebraic Complexity of Quantifier Elimination
, 1996
"... In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of th ..."
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Cited by 201 (29 self)
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In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this 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. Another new feature is that the degrees of the polynomials in the equivalent quantifierfree 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.
Multivariate Polynomials, Duality, and Structured Matrices
 J. of Complexity
, 1999
"... We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlat ..."
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Cited by 51 (29 self)
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We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.
Computing Roadmaps of General SemiAlgebraic Sets
, 1993
"... In this paper we study the problem of determining whether two points lie in the same connected component of a semialgebraic set S. Although we are mostly concerned with sets S # , our algorithm can also decide if points in an arbitrary set S # R can be joined by a semialgebraic path, for any real ..."
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Cited by 50 (2 self)
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In this paper we study the problem of determining whether two points lie in the same connected component of a semialgebraic set S. Although we are mostly concerned with sets S # , our algorithm can also decide if points in an arbitrary set S # R can be joined by a semialgebraic path, for any real closed field R. Our algorithm computes a onedimensional semialgebraic subset ##S# of S (actually of an embedding of S in a space R for a certain real extension field R of the given field R#. ##S# is called the roadmap of S. The basis of this work is the roadmap algorithm described in [3], [4] whichworked only for compact, regularly stratified sets. We measure...
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 37 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Improved Algorithms for Sign Determination and Existential Quantifier Elimination
 The Computer Journal
, 1993
"... Recently there has been a lot of activity in ... In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudopolyno ..."
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Cited by 32 (1 self)
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Recently there has been a lot of activity in ... In this paper we describe a new sign determination method based on the earlier algorithm, but with two advantages: (i) It is faster in the univariate case, and (ii) In the general case, it allows purely symbolic quantifier elimination in pseudopolynomial time. By purely symbolic, we mean that it is possible to eliminate a quantified variable from a system of polynomials no matter what the coefficient values are. The previous methods required the coefficients to be themselves polynomials in other variables
Real Quantifier Elimination in Practice
 Algorithmic Algebra and Number Theory
, 1998
"... We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applicat ..."
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Cited by 32 (5 self)
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We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applications in various fields of science, engineering, and economics.
An algorithm for isolating the real solutions of semialgebraic systems
 J. Symb. Comput
, 2002
"... We propose an algorithm for isolating the real solutions of semialgebraic systems, which has been implemented as a Mapleprogram realzero. The performance of realzero in solving some examples from various applications is presented and the timings are reported. ..."
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Cited by 15 (8 self)
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We propose an algorithm for isolating the real solutions of semialgebraic systems, which has been implemented as a Mapleprogram realzero. The performance of realzero in solving some examples from various applications is presented and the timings are reported.
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
Solving Parametric Polynomial Equations And Inequalities By Symbolic Algorithms
 COMPUTER ALGEBRA IN SCIENCE AND ENGINEERING
, 1995
"... The talk gives a survey on some symbolic algorithmic methods for solving systems of algebraic equations with special emphasis on parametric systems. Besides complex solutions I consider also real solutions of systems including inequalities. The techniques described include the Euclidean algorithm, ..."
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Cited by 12 (1 self)
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The talk gives a survey on some symbolic algorithmic methods for solving systems of algebraic equations with special emphasis on parametric systems. Besides complex solutions I consider also real solutions of systems including inequalities. The techniques described include the Euclidean algorithm, Gröbner bases, characteristic sets, univariate and multivariate SturmSylvester theorems, comprehensive Grobner bases and elimination methods for parametric optimization problems. Some examples illustrate the use of symbolic algorithms for the solution of parametric systems.
Improving conformational searches by geometric screening
 Bioinformatics
"... docking are a timeconsuming process with wide range of applications. Favorable conformations of the ligands that successfully bind with receptors are sought to form stable ligandreceptor complexes. Usually a large number of conformations are generated and their binding energies are examined. We pr ..."
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Cited by 9 (3 self)
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docking are a timeconsuming process with wide range of applications. Favorable conformations of the ligands that successfully bind with receptors are sought to form stable ligandreceptor complexes. Usually a large number of conformations are generated and their binding energies are examined. We propose adding a geometric screening phase before an energy minimization procedure so that only conformations that geometrically fit in the binding site will be prompted for energy calculation. Results: Geometric screening can drastically reduce the number of conformations to be examined from millions (or higher) to thousands (or lower). The method can also handle cases when there are more variables than geometric constraints. An earlystage implementation is able to finish the geometric filtering of conformations for molecules with up to nine variables in one minute. To the best of our knowledge, this is the first time such results are reported deterministically. Contact: