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52
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 75 (34 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 42 (14 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.
On the complexity of the D5 principle
 In Proc. of Transgressive Computing 2006
, 2006
"... The D5 Principle was introduced in 1985 by Jean Della Dora, Claire Dicrescenzo and Dominique Duval in their celebrated note “About a new method for computing in algebraic number fields”. This innovative approach automatizes reasoning based on case discussion and is also known as “Dynamic Evaluation” ..."
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Cited by 29 (17 self)
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The D5 Principle was introduced in 1985 by Jean Della Dora, Claire Dicrescenzo and Dominique Duval in their celebrated note “About a new method for computing in algebraic number fields”. This innovative approach automatizes reasoning based on case discussion and is also known as “Dynamic Evaluation”. Applications of the D5 Principle have been made in Algebra, Computer Algebra, Geometry and Logic. Many algorithms for solving polynomial systems symbolically need to perform standard operations, such as GCD computations, over coefficient rings that are direct products of fields rather than fields. We show in this paper how asymptotically fast algorithms for polynomials over fields can be adapted to this more general context, thanks to the D5 Principle. 1
Fast arithmetic for triangular sets: from theory to practice
 ISSAC'07
, 2007
"... We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, ..."
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Cited by 26 (21 self)
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We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasilinear complexity. The main outcome we have in mind is the acceleration of higherlevel algorithms, by interfacing our lowlevel implementation with languages such as AXIOM or Maple. We show the potential for huge speedups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.
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 25 (17 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.
Triangular Decomposition of SemiAlgebraic Systems
, 2010
"... Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be dec ..."
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Cited by 22 (13 self)
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Regular chains and triangular decompositions are fundamental and welldeveloped tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semialgebraic systems. We show that any such system can be decomposed into finitely many regular semialgebraic systems. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t. the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.
The modpn library: Bringing fast polynomial arithmetic into maple
 IN MICA’08
, 2008
"... We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversi ..."
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Cited by 20 (14 self)
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We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements.
Efficient Implementation of Polynomial Arithmetic in a Multiplelevel Programming Environment
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An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions
"... In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real space. The incrementality comes from the first part of the a ..."
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Cited by 14 (4 self)
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In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real space. The incrementality comes from the first part of the algorithm, where a complex cylindrical tree is constructed by refining a previous complex cylindrical tree with a polynomial constraint. We have implemented our algorithm in Maple. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature. 1