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version 1.2, 2001/10/04 04:12:29 version 1.3, 2001/10/04 08:16:26
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 % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.1 2001/10/03 08:32:58 noro Exp $  % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.2 2001/10/04 04:12:29 noro Exp $
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
   
 \begin{slide}{}  \begin{slide}{}
Line 228  Started Kobe branch [Risa/Asir]
Line 228  Started Kobe branch [Risa/Asir]
 \item OX-RFC102 : communications between servers via MPI  \item OX-RFC102 : communications between servers via MPI
 \end{itemize}  \end{itemize}
   
 \item Rings of differential operators  \item Weyl algebra
   
 \begin{itemize}  \begin{itemize}
 \item Buchberger algorithm [Takayama]  \item Buchberger algorithm [Takayama]
   
 \item $b$-function computation [OT]  \item $b$-function computation [Oaku]
   
 Minimal polynomial computation by modular method  Minimal polynomial computation by modular method
 \end{itemize}  \end{itemize}
Line 290  Modular RUR was comparable with Rouillier's implementa
Line 290  Modular RUR was comparable with Rouillier's implementa
   
 FGb seems much more efficient than our $F_4$ implementation.  FGb seems much more efficient than our $F_4$ implementation.
   
 Singular's Groebner basis computation is also several times  Singular [Singular] is also several times
 faster than Risa/Asir, because Singular seems to have efficient  faster than Risa/Asir, because Singular seems to have efficient
 monomial and polynomial representation.  monomial and polynomial representation.
   
Line 534  Journal of Pure and Applied Algebra (139) 1-3 (1999), 
Line 534  Journal of Pure and Applied Algebra (139) 1-3 (1999), 
 [Hoeij] M. van Heoij, Factoring polynomials and the knapsack problem,  [Hoeij] M. van Heoij, Factoring polynomials and the knapsack problem,
 to appear in Journal of Number Theory (2000).  to appear in Journal of Number Theory (2000).
   
 [SY] T. Shimoyama, K. Yokoyama, Localization and Primary Decomposition of Polynomial Ideals.  J. Symb. Comp. {\bf 22} (1996), 247-277.  
   
 [NY] M. Noro, K. Yokoyama,  [NY] M. Noro, K. Yokoyama,
 A Modular Method to Compute the Rational Univariate  A Modular Method to Compute the Rational Univariate
 Representation of Zero-Dimensional Ideals.  Representation of Zero-Dimensional Ideals.
 J. Symb. Comp. {\bf 28}/1 (1999), 243-263.  J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
   
   [Oaku] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
   local cohomology groups of $D$-modules.
   Advancees in Applied Mathematics, 19 (1997), 61-105.
   \end{slide}
   
   \begin{slide}{}
   
 [OpenMath] {\tt http://www.openmath.org}  [OpenMath] {\tt http://www.openmath.org}
   
 [OpenXM] {\tt http://www.openxm.org}  [OpenXM] {\tt http://www.openxm.org}
Line 553  J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
Line 558  J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
 R\'esolution des syst\`emes z\'ero-dimensionnels.  R\'esolution des syst\`emes z\'ero-dimensionnels.
 Doctoral Thesis(1996), University of Rennes I, France.  Doctoral Thesis(1996), University of Rennes I, France.
   
   [SY] T. Shimoyama, K. Yokoyama, Localization and Primary Decomposition of Polynomial Ideals.  J. Symb. Comp. {\bf 22} (1996), 247-277.
   
   [Singular] {\tt http://www.singular.uni-kl.de}
   
 [Traverso] C. Traverso, \gr trace algorithms. Proc. ISSAC '88 (LNCS 358), 125-138.  [Traverso] C. Traverso, \gr trace algorithms. Proc. ISSAC '88 (LNCS 358), 125-138.
   
 \end{slide}  \end{slide}
Line 645  Guess of a groebner basis by detecting zero reduction 
Line 654  Guess of a groebner basis by detecting zero reduction 
 Homogenization+guess+dehomogenization+check  Homogenization+guess+dehomogenization+check
 \end{itemize}  \end{itemize}
   
 \item Rings of differential operators  \item Weyl Algebra
   
 \begin{itemize}  \begin{itemize}
 \item Groebner basis of a left ideal  \item Groebner basis of a left ideal
Line 730  An ideal whose radical is prime
Line 739  An ideal whose radical is prime
 \begin{slide}{}  \begin{slide}{}
 \fbox{Computation of $b$-function}  \fbox{Computation of $b$-function}
   
 $D$ : the ring of differential operators  $D=K\langle x,\partial \rangle$ : Weyl algebra
   
 $b(s)$ : a polynomial of the smallest degree s.t.  $b(s)$ : a polynomial of the smallest degree s.t.
 there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$  there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$
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