A speed-up of the algorithm for computing comprehensive Gröbner systems

Nabeshima, Katsusuke

doi:10.1145/1277548.1277589

Cited by 39 publications

(27 citation statements)

References 15 publications

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“…In [16], an improved version of the algorithm is reported which removes redundant branches. To reduce the number of branches generated from the SS algorithm, Nabeshima proposed a speed-up algorithm in [12]. The main idea of that algorithm is to exploit disequality parametric constraints for simplification.…”

Section: Rt ≺X If and Only If σā(G) Reduces To 0 Modulo σā(G1) For mentioning

confidence: 99%

“…We have picked examples F3, F5, F6 and F8 from [12] and the examples E4 and E5 from [11]; many other examples can be solved in essentially no time. To get more complex examples, we modified problems from the F5, F6 and F8 in [12] slightly, and they are labeled as S1, S2 and S3.…”

Section: Implementation and Comparative Performancementioning

confidence: 99%

“…Nabeshima's speed-up algorithm [12] improves upon the SS algorithm by using the fact that (i) for every leading power product, only one coefficient needs to be made nonzero, and (ii) Rabinovitch's trick of introducing a new variable can be used to make that polynomial monic. Nabeshima reported that these tricks led to fewer branches of the SSalgorithm for most examples.…”

Section: Introductionmentioning

confidence: 99%

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A new algorithm for computing comprehensive Gröbner systems

Kapur

Sun

Wang

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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A new algorithm for computing a comprehensive Gröbner system of a parametric polynomial ideal over k [U ][X] is presented. This algorithm generates fewer branches (segments) compared to Suzuki and Sato's algorithm as well as Nabeshima's algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröb-ner systems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning's algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbner basis of an ideal over k[U, X] before performing any branches based on parametric constraints. Based on Kalkbrener's results about stability and specialization of Gröbner basis of ideals, the proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröb-ner system, it is only necessary to consider one polynomial for each nondivisible leading power product in k(U ) [X] with the condition that the product of their leading coefficients is not 0; other branches correspond to the cases where this product is 0. In addition, for dealing with a disequality parametric constraint, a probabilistic check is employed for radical membership test of an ideal of parametric constraints. This is in contrast to a general expensive check based on Rabinovitch's trick using a new variable as in Nabeshima's algorithm. The proposed algorithm has been implemented in Magma and experimented with a number of examples from different applications. Its performance (vis a vie number of branches and execution timings) has been compared with the Suzuki-Sato's algorithm and Nabeshima's speed-up algorithm. The algorithm has been successfully used to solve the famous P3P problem from computer vision.

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Section: Rt ≺X If and Only If σā(G) Reduces To 0 Modulo σā(G1) For mentioning

confidence: 99%

Section: Implementation and Comparative Performancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new algorithm for computing comprehensive Gröbner systems

Kapur

Sun

Wang

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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“…example from [16] In table 2 we present the JAS computation times of the examples from Nabeshima. The timings of Nabeshima are the original timings from the article [16].…”

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confidence: 99%

“…The timings of Nabeshima are the original timings from the article [16]. These timings are measured on a Pentium M running at 1.73 GHz.…”

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confidence: 99%

Comprehensive Gröbner Bases in a Java Computer Algebra System

Kredel

2014

Computer Mathematics

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We present an implementation of the algorithms for computing comprehensive Gröbner bases in a Java computer algebra system (JAS). Contrary to approaches to implement comprehensive Gröbner bases with minimal requirements to the computer algebra system, we aim to provide and utilize all necessary algebraic structures occurring in the algorithm. In the implementation of a condition we aim at the maximal semantic exploitation of the occurring algebraic structures: the set of equations equal zero are implemented as an ideal (with Gröbner base computation) and the set of inequalities are implemented as a multiplicative set which is simplified to polynomials of minimal degrees using, for example, square-free decomposition. With our approach we can also make the transition of a comprehensive Gröbner system to a polynomial ring over a (commutative, finite, von Neuman) regular coefficient ring and test or compute Gröbner bases in such polynomial rings.

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Merging Multiple Algorithms for Computing Comprehensive Gröbner Systems Using Parallel Processing

Wada,

Nabeshima

2024

Lecture Notes in Computer Science

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A speed-up of the algorithm for computing comprehensive Gröbner systems

Cited by 39 publications

References 15 publications

A new algorithm for computing comprehensive Gröbner systems

A new algorithm for computing comprehensive Gröbner systems

Comprehensive Gröbner Bases in a Java Computer Algebra System

Merging Multiple Algorithms for Computing Comprehensive Gröbner Systems Using Parallel Processing

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