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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  FGb seems much more efficient than our $F_4$ implement
Line 290  FGb seems much more efficient than our $F_4$ implement
   
 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  to appear in Journal of Number Theory (2000).
Line 534  to appear in Journal of Number Theory (2000).
 [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  Doctoral Thesis(1996), University of Rennes I, France.
Line 558  Doctoral Thesis(1996), University of Rennes I, France.
 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  Homogenization+guess+dehomogenization+check
Line 654  Homogenization+guess+dehomogenization+check
 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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