2007 Fiscal Year Final Research Report Summary
Development of efficient algorithms and software for solving parametric systems
| Project/Area Number |
17340028
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
NORO Masayuki Kobe University, Graduate School of Science, Professor (50332755)
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| Co-Investigator(Kenkyū-buntansha) |
TAKAYAMA Nobuki Kobe University, Graduate School of Science, Professor (30188099)
SUZUKI Akira Kobe University, Administrative Office, Assistant Professor (50330519)
YOKOYAMA Kazuhiro Rikkyo University, Faculty of Science, Professor (30333454)
SATO Yosuke Tokyo University of Science, Faculty of Science, Professor (50257820)
OHARA Katsuyoshi Kanazawa University, Faculty of Science, Assistant Professor (00313635)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Groebner Basis / F4 algorithm / Algebraic extension / CGB / Mathematical Software / Hvnereeometric equation / polynomial system / modular computation |
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| Research Abstract |
The purpose of this research is to develop efficient algorithms and software for solving polynomial systems or system of differential equations containing parameters. The research results are as follows:
1. Dynamic evaluation allows us to define algebraic numbers by non-irreducible defining polynomial. We reformulated the dynamic evaluation by using the notion of ideal quotient, and we proposed a modular method to improve the computation. We applied the new method for the computation of discrete comprehensive Groebner basis.
2. We proposed a new method for computing comprehensive Groebner basis (CGB) and comprehensive Groebner system (CGS). This method can be implemented by using Groebner basis computation over usual polynomial ring and it improves the computation of CGB or CGS.
3. We investigated the structures of polynomial ideals with parameters. In particular, we focused on the stability of parametric polynomial ideals and we found a periodicity and asymptotic behavior of such ideals in simple cases.
4. We proposed algorithms for doing division with remainder in rings of differential operators over the field of rational functions or the power series rings. We constructed convergent solutions of A-hypergeometric systems. By using the system of difference equations for Gauss hypergeometric functions, we derived their quadratic relations. For computing these objects we developed a software package named ‘yang' for computing in rings of difference-differential operators over the field of rational functions.
5. All the results above have been implemented in a computer algebra system Risa/Asir, which is available from our web site. In the conferences of Japan Mathematical Society, we held workshops named "Mathematical software and free documents". Furthermore, we constructed a virtual machine on which KNOPPIX/Math runs and we distributed DVDs containing the virtual machine.
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