On the Parallel Computation of Boolean Gr¨obner Bases Shutaro Inoue & Yosuke Sato A commutative ring with an identity such that every element of which is idempotent is called a boolean ring. For a boolean ring B, a quotient ring B[X1 , . . . , Xn ]/〈X12 − X1 , . . . , Xn2 − Xn 〉 also becomes a boolean ring. It is called a boolean polynomial ring. A Gr¨ obner basis in a boolean polynomial ring (called a boolean Gr¨ obner basis) is first introduced in [2] and further properties are studied in [3] and [5]. A naive approach for the parallel computation of boolean Gr¨obner bases is also discussed in [4]. In the talk, we introduce a more sophisticated parallel algorithm for the computation of boolean Gr¨ obner bases, which is based on the results of [5]. We show our algorithm is satisfactorily efficient using benchmark data of our computation experiments obtained by our parallel implementation of the algorithm on the computer algebra system Risa/Asir([1]) with the parallel computation environment OpenXM([6]).

References [1] Noro, M. and Takeshima, T. Risa/Asir - Computer Algebra System. International Symposium on Symbolic and Algebraic Computation(ISSAC 92),Proceedings, 387396, ACM Press, 1992. [2] Sakai, K. and Sato, Y. Boolean Gr¨obner bases.ICOT Technical Memorandum 488. http://www.icot.or.jp/ARCHIVE/Museum/TRTM/tm-list-E.html [3] Sato, Y. A new type of canonical Gr¨obner Bases in polynomial rings over Von Neumann regular rings, International Symposium on Symbolic and Algebraic Computation(ISSAC 98),Proceedings, pp 317-321, ACM Press, 1998. [4] Sato, Y. and Suzuki, A. Parallel computation of Boolean Gr¨obner Bases. Proceedings of the Fourth Asian Technology Conference in Mathematics(ATCM 1999), pp 265274, 1999. [5] Sato, Y. Nagai, A and Inoue, S. On the Computation of Elimination Ideals of Boolean Polynomial Rings. The Asian Symposium on Computer Mathematics(ASCM 2007). To appear in LNCS Vol.5081. [6] Takayama, N. etc. Open message eXchage protocol for Mathematics,        http://www.math.kobe-u.ac.jp/OpenXM/

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On the Parallel Computation of Boolean Gröbner Bases

obtained by our parallel implementation of the algorithm on the computer algebra system. Risa/Asir([1]) with the parallel computation environment OpenXM([6]).
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