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26
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.
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.
Computations modulo regular chains
 In Proc. of ISSAC’09
, 2009
"... The computation of triangular decompositions involves two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations based on modular methods and fast polynomial arithmetic. We rely on new results con ..."
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Cited by 17 (9 self)
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The computation of triangular decompositions involves two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations based on modular methods and fast polynomial arithmetic. We rely on new results connecting polynomial subresultants and GCDs modulo regular chains. We report on extensive experimentation, comparing our code to preexisting Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.
FFTbased dense polynomial arithmetic on multicores
 Proc. of HPCS’09
, 2009
"... Abstract. We report efficient implementation techniques for FFTbased dense multivariate polynomial arithmetic over finite fields, targeting multicores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ f ..."
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Cited by 10 (5 self)
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Abstract. We report efficient implementation techniques for FFTbased dense multivariate polynomial arithmetic over finite fields, targeting multicores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ for polynomial arithmetic such as normal form computation. Since bivariate multiplication applied to balanced data is a good kernel for these routines, we provide an indepth study on the performance and the cutoff criteria of our different implementations for this operation. We also show that, not only optimized parallel multiplication can improve the performance of higherlevel algorithms such as normal form computation but also this composition is necessary for parallel normal form computation to reach peak performance on a variety of problems that we have tested.
Balanced Dense Polynomial Multiplication on Multicores
"... Abstract — In symbolic computation, polynomial multiplication is a fundamental operation akin to matrix multiplication in numerical computation. We present efficient implementation strategies for FFTbased dense polynomial multiplication targeting multicores. We show that balanced input data can ma ..."
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Cited by 8 (4 self)
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Abstract — In symbolic computation, polynomial multiplication is a fundamental operation akin to matrix multiplication in numerical computation. We present efficient implementation strategies for FFTbased dense polynomial multiplication targeting multicores. We show that balanced input data can maximize parallel speedup and minimize cache complexity for bivariate multiplication. However, unbalanced input data, which are common in symbolic computation, are challenging. We provide efficient techniques, what we call contraction and extension, to reduce multivariate (and univariate) multiplication to balanced bivariate multiplication. Our implementation in Cilk++ demonstrates good speedup on multicores. Keywords parallel symbolic computation; parallel polynomial multiplication; parallel multidimensional FFT/TFT; Cilk++; multicore; I.
Highperformance symbolic computation in a hybrid compiledinterpreted programming environment
 In ICCSA’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 4 (4 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. 1
Fast Arithmetics in ArtinSchreier Towers over Finite Fields
"... An ArtinSchreier tower over the finite field Fp is a tower of field extensions generated by polynomials of the form X p − X − α. Following Cantor and Couveignes, we give algorithms with quasilinear time complexity for arithmetic operations in such towers. As an application, we present an implement ..."
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Cited by 3 (2 self)
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An ArtinSchreier tower over the finite field Fp is a tower of field extensions generated by polynomials of the form X p − X − α. Following Cantor and Couveignes, we give algorithms with quasilinear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes ’ algorithm for computing isogenies between elliptic curves using the ptorsion. AlgoCategories and Subject Descriptors: