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On the Theories of Triangular Sets
 J. SYMB. COMP
, 1999
"... Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent. ..."
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Cited by 116 (36 self)
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Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.
On triangular decompositions of algebraic varieties
 Presented at the MEGA2000 Conference
, 1999
"... We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifti ..."
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Cited by 81 (35 self)
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We propose an efficient algorithm for computing triangular decompositions of algebraic varieties. It is based on an incremental process and produces components in order of decreasing dimension. The combination of these two major features is obtained by means of lazy evaluation techniques and a lifting property for calculations modulo regular chains. This allows a good management of the intermediate computations, as confirmed by several implementations and applications of this work. Our algorithm is also well suited for parallel execution.
Computing Cylindrical Algebraic Decomposition via Triangular Decomposition
, 2009
"... Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensiv ..."
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Cited by 51 (18 self)
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Cylindrical algebraic decomposition is one of the most important tools for computing with semialgebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensive triangular decomposition in order to obtain an Finvariant cylindrical decomposition of the ndimensional complex space, from which we extract an Finvariant cylindrical algebraic decomposition of the ndimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.
Algorithms for Computing Triangular Decomposition of Polynomial Systems
, 2011
"... We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we reg ..."
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Cited by 29 (21 self)
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We discuss algorithmic advances which have extended the pioneer work of Wu on triangular decompositions. We start with an overview of the key ideas which have led to either better implementation techniques or a better understanding of the underlying theory. We then present new techniques that we regard as essential to the recent success and for future research directions in the development of triangular decomposition methods.
Properness defects of projections and computation of one point in each connected component of a real algebraic set
, 2003
"... ..."
Efficient computations of irredundant triangular decompositions with the regularchains library
 Proc. of CASA2007
, 2007
"... Abstract. We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for remo ..."
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Cited by 13 (9 self)
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Abstract. We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for removing redundant components can be reduced to a small portion of the total time needed for solving these systems. Since testing the inclusion between two quasicomponents can be as expensive as a radical membership test, and many pairs of quasicomponents may need to be compared, we believe that we have obtained an efficient solution.
Properness defects of projections and computation of at least one point in each connected component of a real algebraic set
, 2004
"... ..."
Efficient Algorithms for the k
 Maximum Sums, ISAAC 2004, LNCS 3341
"... We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a weakened notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the subalgorithms. Extract ..."
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Cited by 11 (0 self)
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We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a weakened notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the subalgorithms. Extracting common work from similar expensive computations is also a key feature of our algorithms. In our experimental results the implementation of our new algorithms, realized with the RegularChains library in Maple, outperforms solvers with similar specifications by several orders of magnitude on sufficiently difficult problems. Categories and Subject Descriptors
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, (CADbased) 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, (CADbased) 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 selfassembling micelle systems with chemical sinks are analyzed in detail. We provide experimental results with comparisons for 15 biological models taken from the literature.