PGB is a software package for computing parametric Gröbner bases and related objects in several domains. It is implemented in the computer algebra system Risa/Asir by Katsusuke Nabeshima. ...

# Dr. Katsusuke Nabeshima

### Research Area

Computer Algebra## Software

Authors: Katsusuke Nabeshima

More## Publications

### 2007

[Nabeshima]

### Comprehensive Groebner bases in various domains

#### K. Nabeshima

RISC-Linz. PhD Thesis. April 2007. [ps] [pdf]@

author = {K. Nabeshima},

title = {{Comprehensive Groebner bases in various domains}},

language = {english},

year = {2007},

month = {April},

translation = {0},

school = {RISC-Linz},

length = {194}

}

**phdthesis**{RISC3116,author = {K. Nabeshima},

title = {{Comprehensive Groebner bases in various domains}},

language = {english},

year = {2007},

month = {April},

translation = {0},

school = {RISC-Linz},

length = {194}

}

[Nabeshima]

### A Speed-Up Algorithm for Computing Comprehensive Groebner Systems

#### K. Nabeshima

In: ISSAC 2007, C. W. Brown (ed.), pp. 299-306. 2007. ACM-press, 978-1-59593-743-8. [ps] [pdf]@

author = {K. Nabeshima},

title = {{A Speed-Up Algorithm for Computing Comprehensive Groebner Systems}},

booktitle = {{ISSAC 2007}},

language = {english},

abstract = {We introduce a new algorithm for computing comprehensive Groebner systems. There exists the Suzuki-Sato algorithm for computing comprehensive Groebner systems. The Suzuki-Sato algorithm often creates overmuch cells of the parameter space for comprehensive Groebner systems. Therefore the computation becomes heavy. However, by using inequations (``not equal zero''), we can obtain different cells. In many cases, this number of cells of parameter space is smaller than that of Suzuki-Sato's. Therefore, our new algorithm is more efficient than Suzuki-Sato's one, and outputs a nice comprehensive Groebner system. Our new algorithm has been implemented in the computer algebra system Risa/Asir. We compare the runtime of our implementation with the Suzuki-Sato algorithm and find our algorithm superior in many cases.},

pages = {299--306},

publisher = {ACM-press},

isbn_issn = {978-1-59593-743-8},

year = {2007},

editor = {C. W. Brown},

refereed = {yes},

length = {8}

}

**inproceedings**{RISC3154,author = {K. Nabeshima},

title = {{A Speed-Up Algorithm for Computing Comprehensive Groebner Systems}},

booktitle = {{ISSAC 2007}},

language = {english},

abstract = {We introduce a new algorithm for computing comprehensive Groebner systems. There exists the Suzuki-Sato algorithm for computing comprehensive Groebner systems. The Suzuki-Sato algorithm often creates overmuch cells of the parameter space for comprehensive Groebner systems. Therefore the computation becomes heavy. However, by using inequations (``not equal zero''), we can obtain different cells. In many cases, this number of cells of parameter space is smaller than that of Suzuki-Sato's. Therefore, our new algorithm is more efficient than Suzuki-Sato's one, and outputs a nice comprehensive Groebner system. Our new algorithm has been implemented in the computer algebra system Risa/Asir. We compare the runtime of our implementation with the Suzuki-Sato algorithm and find our algorithm superior in many cases.},

pages = {299--306},

publisher = {ACM-press},

isbn_issn = {978-1-59593-743-8},

year = {2007},

editor = {C. W. Brown},

refereed = {yes},

length = {8}

}

[Nabeshima]

### Reduced Groebner Bases in Polynomial Rings over a Polynomial Ring

#### K. Nabeshima

Mathematics in Computer Science 1(2), pp. ??-??. 2007. Brikhauser/Springer, to appear. [pdf] [ps]@

author = {K. Nabeshima},

title = {{Reduced Groebner Bases in Polynomial Rings over a Polynomial Ring}},

language = {english},

abstract = {We define reduced Gr\"obner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gr\"obner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Groebner bases by these algorithms. In this paper we propose a new notion of reduced Groebner bases in polynomial rings over a polynomial ring and we show that every ideal has a unique reduced Groebner basis.},

journal = {Mathematics in Computer Science},

volume = {1},

number = {2},

pages = {??--??},

publisher = {Brikhauser/Springer},

isbn_issn = {to appear},

year = {2007},

refereed = {yes},

length = {17}

}

**article**{RISC3156,author = {K. Nabeshima},

title = {{Reduced Groebner Bases in Polynomial Rings over a Polynomial Ring}},

language = {english},

abstract = {We define reduced Gr\"obner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gr\"obner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Groebner bases by these algorithms. In this paper we propose a new notion of reduced Groebner bases in polynomial rings over a polynomial ring and we show that every ideal has a unique reduced Groebner basis.},

journal = {Mathematics in Computer Science},

volume = {1},

number = {2},

pages = {??--??},

publisher = {Brikhauser/Springer},

isbn_issn = {to appear},

year = {2007},

refereed = {yes},

length = {17}

}

### 2006

[Nabeshima]

### Reduced Groebner bases in polynomial rings over a polynomial ring

#### K. Nabeshima

In: Internatial Conference on Mathematical Aspects of Computer and Information Sciences, Wang, D. and Zheng, Z. (ed.), pp. 15-32. 2006. ----.@

author = {K. Nabeshima},

title = {{Reduced Groebner bases in polynomial rings over a polynomial ring}},

booktitle = {{Internatial Conference on Mathematical Aspects of Computer and Information Sciences}},

language = {english},

abstract = {We define reduced Gr\"obner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gr\"obner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Gr\"obner bases by these algorithms. In this paper we give problems for computing the reduced Gr\"obner bases and show how to solve these problems.},

pages = {15--32},

isbn_issn = {--------},

year = {2006},

editor = {Wang and D. and Zheng and Z.},

refereed = {yes},

length = {18}

}

**inproceedings**{RISC2997,author = {K. Nabeshima},

title = {{Reduced Groebner bases in polynomial rings over a polynomial ring}},

booktitle = {{Internatial Conference on Mathematical Aspects of Computer and Information Sciences}},

language = {english},

abstract = {We define reduced Gr\"obner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gr\"obner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Gr\"obner bases by these algorithms. In this paper we give problems for computing the reduced Gr\"obner bases and show how to solve these problems.},

pages = {15--32},

isbn_issn = {--------},

year = {2006},

editor = {Wang and D. and Zheng and Z.},

refereed = {yes},

length = {18}

}

### 2005

[Nabeshima]

### A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields

#### K. Nabeshima

In: 7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05), Petcu, D. (ed.), Proceedings of 7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05), pp. 10-17. 2005. to appear in IEEE Press.@

author = {K. Nabeshima},

title = {{A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields}},

booktitle = {{7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05)}},

language = {english},

abstract = {In this paper we describe comprehensive Gröbner bases over finite fields by direct product of fields. In general, representations of comprehensive Gröbner bases have some conditions on parameters. However, in finite fields we can construct comprehensive Gröbner bases without conditions by the theory of von Neumann regular rings. Our comprehensive Gröbner bases are defined as Gröbner bases in polynomial rings over commutative von Neumann regular rings, hence our comprehensive Gröbner bases have some nice properties. Our method is different from the methods of Weispfenning (CGB,CCGB), Montes (DisPGB), Sato and Suzuki (ACGB). },

pages = {10--17},

year = {2005},

note = {to appear in IEEE Press},

editor = {Petcu and D.},

refereed = {yes},

length = {8},

conferencename = {7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05)}

}

**inproceedings**{RISC2797,author = {K. Nabeshima},

title = {{A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields}},

booktitle = {{7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05)}},

language = {english},

abstract = {In this paper we describe comprehensive Gröbner bases over finite fields by direct product of fields. In general, representations of comprehensive Gröbner bases have some conditions on parameters. However, in finite fields we can construct comprehensive Gröbner bases without conditions by the theory of von Neumann regular rings. Our comprehensive Gröbner bases are defined as Gröbner bases in polynomial rings over commutative von Neumann regular rings, hence our comprehensive Gröbner bases have some nice properties. Our method is different from the methods of Weispfenning (CGB,CCGB), Montes (DisPGB), Sato and Suzuki (ACGB). },

pages = {10--17},

year = {2005},

note = {to appear in IEEE Press},

editor = {Petcu and D.},

refereed = {yes},

length = {8},

conferencename = {7th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC05)}

}

[Nabeshima]

### A computation method for ACGB-V

#### K. Nabeshima

In: Algorithm Algebra and Logic (A3L) 2005, Dolzmann, A., Seidl, S., and Sturm, T. (ed.), Proceedings of Algorithm Algebra and Logic 2005, Conference in Honor of the 60th Birthday of Volker Weispfenning, pp. 173-180. April 2005. BOD Norderstedt, ISBN 3-8334-2669-1. [url]@

author = {K. Nabeshima},

title = {{A computation method for ACGB-V}},

booktitle = {{Algorithm Algebra and Logic (A3L) 2005}},

language = {english},

abstract = {In this paper we describe an algorithm for ACGB-V (Alternative Comprehensive Groebner Basis on Varieties). Discrete comprehensive Groebner bases were proposed by Sato, Suzuki and Nabeshima. Discrete comprehensive Groebner bases are a special type of ACGB-V. In this paper we extend the method of discrete comprehensive Groebner basesto the general method of ACGB-V which uses Weispfenning's theory of Grobner bases in polynomial rings over commutative von Neumann regualr rings.},

pages = {173--180},

publisher = {BOD Norderstedt},

isbn_issn = {ISBN 3-8334-2669-1},

year = {2005},

month = {April},

editor = {Dolzmann and A. and Seidl and S. and and Sturm and T.},

refereed = {yes},

length = {8},

conferencename = {Algorithm Algebra and Logic 2005, Conference in Honor of the 60th Birthday of Volker Weispfenning},

url = {http://www.a3l.org/}

}

**inproceedings**{RISC2792,author = {K. Nabeshima},

title = {{A computation method for ACGB-V}},

booktitle = {{Algorithm Algebra and Logic (A3L) 2005}},

language = {english},

abstract = {In this paper we describe an algorithm for ACGB-V (Alternative Comprehensive Groebner Basis on Varieties). Discrete comprehensive Groebner bases were proposed by Sato, Suzuki and Nabeshima. Discrete comprehensive Groebner bases are a special type of ACGB-V. In this paper we extend the method of discrete comprehensive Groebner basesto the general method of ACGB-V which uses Weispfenning's theory of Grobner bases in polynomial rings over commutative von Neumann regualr rings.},

pages = {173--180},

publisher = {BOD Norderstedt},

isbn_issn = {ISBN 3-8334-2669-1},

year = {2005},

month = {April},

editor = {Dolzmann and A. and Seidl and S. and and Sturm and T.},

refereed = {yes},

length = {8},

conferencename = {Algorithm Algebra and Logic 2005, Conference in Honor of the 60th Birthday of Volker Weispfenning},

url = {http://www.a3l.org/}

}

[Nabeshima]

### A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields

#### K. Nabeshima

In: ACA 2005, Shirayanagi, K. (ed.), Proceedings of Conference on Applications of Computer Algebra, pp. 54-55. August 2005. 4-903027-02-3. [url]@

author = {K. Nabeshima},

title = {{A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields}},

booktitle = {{ACA 2005}},

language = {english},

abstract = {We describe comprehensive Gröbner bases over finite fields by direct product of fields. In general, representatioins of comprehensive Gröbner bases have some conditions on parameters. However, in finite fields we can construct comprehensive Gröbner bases without conditions by the theory of von Neumann regular rings .\\Alternative comprehensive Gröbner bases (ACGB) are also bases on the theory of von Neumann regular rings. However, ACGB are defined for infinite fields, we can not use the method given by ACGB for finite fields.The comprehensive Gröbner bases we are to describe are defined as Gröbner bases in polynomial rings over commutative von Neumann regular rings, hence the comprehensive Gröbner bases have some nice properties which we also describe. },

pages = {54--55},

isbn_issn = {4-903027-02-3},

year = {2005},

month = {August},

editor = {Shirayanagi and K.},

refereed = {no},

length = {2},

conferencename = {Conference on Applications of Computer Algebra},

url = {http://www.jssac.org/Conference/ACA/}

}

**inproceedings**{RISC2794,author = {K. Nabeshima},

title = {{A Direct Products of Fields Approach to Comprehensive Gröbner Bases over Finite Fields}},

booktitle = {{ACA 2005}},

language = {english},

abstract = {We describe comprehensive Gröbner bases over finite fields by direct product of fields. In general, representatioins of comprehensive Gröbner bases have some conditions on parameters. However, in finite fields we can construct comprehensive Gröbner bases without conditions by the theory of von Neumann regular rings .\\Alternative comprehensive Gröbner bases (ACGB) are also bases on the theory of von Neumann regular rings. However, ACGB are defined for infinite fields, we can not use the method given by ACGB for finite fields.The comprehensive Gröbner bases we are to describe are defined as Gröbner bases in polynomial rings over commutative von Neumann regular rings, hence the comprehensive Gröbner bases have some nice properties which we also describe. },

pages = {54--55},

isbn_issn = {4-903027-02-3},

year = {2005},

month = {August},

editor = {Shirayanagi and K.},

refereed = {no},

length = {2},

conferencename = {Conference on Applications of Computer Algebra},

url = {http://www.jssac.org/Conference/ACA/}

}