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45
The Complexity of Quantifier Elimination and Cylindrical Algebraic Decomposition
- Proceedings ISSAC 2007
, 2007
"... This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simp ..."
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Cited by 41 (18 self)
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This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction — roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worst-case running times of several modern quantifier elimination algorithms and CAD-based quantifier elimination when the number of quantifier alternations is constant.
Finding all real points of a complex curve
, 2006
"... An algorithm is given to compute the real points of the irreducible one-dimensional complex components of the solution sets of systems of polynomials with real coefficients. The algorithm is based on homo-topy continuation and the numerical irreducible decomposition. An extended application is made ..."
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Cited by 21 (9 self)
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An algorithm is given to compute the real points of the irreducible one-dimensional complex components of the solution sets of systems of polynomials with real coefficients. The algorithm is based on homo-topy continuation and the numerical irreducible decomposition. An extended application is made to Griffis-Duffy platforms, a class of Stewart-Gough platform robots. 2000 Mathematics Subject Classification. Primary 65H10; Sec-ondary 65H20, 14Q99. Key words and phrases. Homotopy continuation, numerical alge-braic geometry, real polynomial systems. In this article we give a numerical algorithm to find the real zero- and
Testing sign conditions on a multivariate polynomial and applications
- MATHEMATICS IN COMPUTER SCIENCE
"... Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each conne ..."
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Cited by 17 (8 self)
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Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f − e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set K0(f) of critical values of the mapping f and K∞(f) of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semi-algebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach.
Algebraic approaches to stability analysis of biological systems
- MATH. COMPUT. SCI
, 2008
"... In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gröbner bases, (CAD-based) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the st ..."
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Cited by 11 (1 self)
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In this paper, we improve and extend the approach of Wang and Xia for stability analysis of biological systems by making use of Gröbner bases, (CAD-based) quantifier elimination, and discriminant varieties, as well as the stability criterion of Liénard and Chipart, and showing how to analyze the stability of Hopf bifurcation points. The stability and bifurcations for a class of self-assembling micelle systems with chemical sinks are analyzed in detail. We provide experimental results with comparisons for 15 biological models taken from the literature.
Classification of the Perspective-Three-Point Problem, Discriminant Variety and Real Solving Polynomial Systems of Inequalities
, 2008
"... Classifying the Perspective-Three-Point problem (abbreviated by P3P in the sequel) consists in determining the number of possible positions of a camera with respect to the apparent position of three points. In the case where the three points form an isosceles triangle, we give a full classification ..."
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Cited by 4 (0 self)
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Classifying the Perspective-Three-Point problem (abbreviated by P3P in the sequel) consists in determining the number of possible positions of a camera with respect to the apparent position of three points. In the case where the three points form an isosceles triangle, we give a full classification of the P3P. This leads to consider a polynomial system of polynomial equations and inequalities with 4 parameters which is generically zero-dimensional. In the present situation, the parameters represent the apparent position of the three points so that solving the problem means determining all the possible numbers of real solutions with respect to the parameters’ values and give a sample point for each of these possible numbers. One way for solving such systems consists first in computing a discriminant variety. Then, one has to
Sweeping Algebraic Curves for Singular Solutions
, 2009
"... Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when ..."
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Cited by 3 (1 self)
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Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.
Cylinders through five points: complex and real enumerative geometry
- the Sixth International Workshop on Automated Deduction in Geometry (ADG
, 2006
"... Abstract. It is known that five points in R 3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspo ..."
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Cited by 3 (1 self)
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Abstract. It is known that five points in R 3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in R 3. We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real valued solution cardinalities.
