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version 1.6, 2001/10/09 11:44:43 version 1.7, 2001/10/10 06:32:10
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 % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.5 2001/10/09 01:44:21 noro Exp $  % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.6 2001/10/09 11:44:43 noro Exp $
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
   
 \begin{slide}{}  \begin{slide}{}
Line 8 
Line 8 
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
   \fbox{Integration of mathematical software systems}
   
   \begin{itemize}
   \item Data integration
   
   \begin{itemize}
   \item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]
   \end{itemize}
   
   Standards for representing mathematical objects
   
   \item Control integration
   
   \begin{itemize}
   \item MCP [WANG99], OMEI [LIAO01]
   \end{itemize}
   
   Protocols for remote subroutine calls or session management
   
   \item Combination of two integrations
   
   \begin{itemize}
   \item MathLink, OpenMath+MCP, MP+MCP
   
   and OpenXM ({\tt http://www.openxm.org})
   \end{itemize}
   
   Both are necessary for practical implementation
   
   \end{itemize}
   \end{slide}
   \begin{slide}{}
 \fbox{OpenXM (Open message eXchange protocol for Mathematics) }  \fbox{OpenXM (Open message eXchange protocol for Mathematics) }
   
 \begin{itemize}  \begin{itemize}
Line 27  Encoding : CMO (Common Mathematical Object format)
Line 59  Encoding : CMO (Common Mathematical Object format)
   
 Serialized representation of mathematical object  Serialized representation of mathematical object
   
 --- Main idea was borrowed from OpenMath [OpenMath]  --- Main idea was borrowed from OpenMath
 \item Command  \item Command
   
 stack machine command --- server is a stackmachine  stack machine command --- server is a stackmachine
Line 36  stack machine command --- server is a stackmachine
Line 68  stack machine command --- server is a stackmachine
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{OpenXM and OpenMath}  \fbox{A computer algebra system Risa/Asir}
   
 \begin{itemize}  ({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
 \item OpenMath  
   
 \begin{itemize}  \begin{itemize}
 \item A standard for representing mathematical objects  \item Traditional style software for polynomial computation
   
 \item CD (Content Dictionary) : assigns semantics to symbols  
   
 \item Phrasebook : convesion between internal and OpenMath objects.  
   
 \item Encoding : format for actual data exchange  
 \end{itemize}  
   
 \item OpenXM  
   
 \begin{itemize}  
 \item Specification for encoding and exchanging messages  
   
 \item It also specifies behavior of servers and session management  
 \end{itemize}  
   
 \end{itemize}  
 \end{slide}  
 \begin{slide}{}  
 \fbox{A computer algebra system Risa/Asir}  
   
 \begin{itemize}  
 \item Old style software for polynomial computation  
   
 No domain specification, automatic expansion  No domain specification, automatic expansion
   
 \item User language with C-like syntax  \item User language with C-like syntax
Line 86  Whole source tree is available via CVS
Line 93  Whole source tree is available via CVS
 \begin{itemize}  \begin{itemize}
 \item Risa/Asir is a main client in OpenXM package.  \item Risa/Asir is a main client in OpenXM package.
 \item An OpenXM server {\tt ox\_asir}  \item An OpenXM server {\tt ox\_asir}
 \item An library with OpemXM library inteface {\tt libasir.a}  \item A library with OpenXM library interface {\tt libasir.a}
 \end{itemize}  \end{itemize}
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Aim of developing Risa/Asir}  \fbox{Goal of developing Risa/Asir}
   
 \begin{itemize}  \begin{itemize}
 \item Efficient implementation in specific area  \item Efficient implementation in specific area
   
 Polynomial factorization, Groebner basis related computation  \begin{itemize}
   \item Polynomial factorization
   
 $\Rightarrow$ my main motivation  \item Groebner basis related computation
   
 \item Front-end of a general purpose math software  Main target : coefficient swells in characteristic 0 cases
   
 Risa/Asir contains PARI library [PARI] from the very beginning  Main tool : modular method
   \end{itemize}
   
 It also acts as a main client of OpenXM package  \item Front-end or server of a general purpose math software
   
   We do not persist in self-containedness
   
   \begin{itemize}
   
   \item contains PARI library ({\tt http://www.parigp-home.de}) from the very beginning
   
   \item also acts as a main client of OpenXM package
   
   One can use various OpenXM servers
   
 \end{itemize}  \end{itemize}
   
   \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
Line 116  It also acts as a main client of OpenXM package
Line 137  It also acts as a main client of OpenXM package
 \begin{itemize}  \begin{itemize}
 \item Fundamental polynomial arithmetics  \item Fundamental polynomial arithmetics
   
 recursive representaion and distributed representation  recursive representation and distributed representation
   
 \item Polynomial factorization  \item Polynomial factorization
   
Line 129  recursive representaion and distributed representation
Line 150  recursive representaion and distributed representation
 \item Groebner basis computation  \item Groebner basis computation
   
 \begin{itemize}  \begin{itemize}
 \item Buchberger and $F_4$ [Faug\'ere] algorithm  \item Buchberger and $F_4$ [FAUG99] algorithm
   
 \item Change of ordering/RUR [Rouillier] of 0-dimensional ideals  \item Change of ordering/RUR [ROUI96] of 0-dimensional ideals
   
 \item Primary ideal decomposition  \item Primary ideal decomposition
   
Line 146  recursive representaion and distributed representation
Line 167  recursive representaion and distributed representation
 \begin{itemize}  \begin{itemize}
 \item 1989  \item 1989
   
 Start of Risa/Asir with Boehm's conservative GC [Boehm]  Start of Risa/Asir with Boehm's conservative GC
   
   ({\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
   
 \item 1989-1992  \item 1989-1992
   
 Univariate and multivariate factorizers over {\bf Q}  Univariate and multivariate factorizers over {\bf Q}
Line 174  Multivariate factorization over small finite fields (i
Line 197  Multivariate factorization over small finite fields (i
 \begin{itemize}  \begin{itemize}
 \item 1992-1994  \item 1992-1994
   
 User language $\Rightarrow$ C version; trace lifting [Traverso]  User language $\Rightarrow$ C version; trace lifting [TRAV88]
   
 \item 1994-1996  \item 1994-1996
   
 Trace lifting with homogenization  Trace lifting with homogenization
   
 Omitting GB check by compatible prime [NY]  Omitting GB check by compatible prime [NOYO99]
   
 Modular change of ordering/RUR [NY]  Modular change of ordering/RUR [NOYO99]
   
 Primary ideal decompositon [SY]  Primary ideal decomposition [SHYO96]
   
 \item 1996-1998  \item 1996-1998
   
 Effifcient content reduction during NF computation and its parallelization  Efficient content reduction during NF computation [NORO97]
 [Noro] (Solved {\it McKay} system for the first time)  Solved {\it McKay} system for the first time
   
 \item 1998-2000  \item 1998-2000
   
Line 197  Test implementation of $F_4$
Line 220  Test implementation of $F_4$
   
 \item 2000-current  \item 2000-current
   
 Buchberger algorithm in Weyl algebra [Takayama]  Buchberger algorithm in Weyl algebra [TAKA90]
   
 Efficient $b$-function computation by a modular method  Efficient $b$-function computation by a modular method
 \end{itemize}  \end{itemize}
Line 212  Efficient $b$-function computation by a modular method
Line 235  Efficient $b$-function computation by a modular method
 Over {\bf Q} : fine compared with existing software  Over {\bf Q} : fine compared with existing software
 like REDUCE, Mathematica, maple  like REDUCE, Mathematica, maple
   
 Univarate, over algebraic number fields :  Univariate, over algebraic number fields :
 fine because of some tricks for polynomials  fine because of some tricks for polynomials
 derived from norms.  derived from norms.
   
Line 220  derived from norms.
Line 243  derived from norms.
   
 Multivariate : moderate  Multivariate : moderate
   
 Univariate : completely obsolete by M. van Hoeij's new algorithm  Univariate : completely obsoleted by M. van Hoeij's new algorithm
 [Hoeij]  [HOEI00]
 \end{itemize}  \end{itemize}
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
   \fbox{Timing data --- Factorization}
   
   \underline{Univariate; over {\bf Q}}
   
   $N_i$ : a norm of a poly, $\deg(N_i) = i$
   \begin{center}
   \begin{tabular}{|c||c|c|c|c|} \hline
                   & $N_{105}$ & $N_{120}$ & $N_{168}$ & $N_{210}$ \\ \hline
   Asir    & 0.86  & 59 & 840 & hard \\ \hline
   Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline
   Singular& hard? & hard?& hard? & hard? \\ \hline
   CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline
   NTL-5.2 & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline
   \end{tabular}
   \end{center}
   
   \underline{Multivariate; over {\bf Q}}
   
   $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}
   \begin{center}
   \begin{tabular}{|c||c|c|c|c|c|} \hline
           & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
   Asir    & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline
   Singular& $>$15min      & ---   & ---& ---& ---\\ \hline
   CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline
   Mathematica& 0.2        & 16    & 23 & 36 & 1.1 \\ \hline
   \end{tabular}
   \end{center}
   
   --- : not tested
   \end{slide}
   \begin{slide}{}
 \fbox{Performance --- Groebner basis related computation}  \fbox{Performance --- Groebner basis related computation}
   
 \begin{itemize}  \begin{itemize}
Line 242  Modular RUR was comparable with Rouillier's implementa
Line 297  Modular RUR was comparable with Rouillier's implementa
   
 DRL basis of {\it McKay}:  DRL basis of {\it McKay}:
   
 5 days on Risa/Asir, 53 seconds on Faugere FGb  5 days on Risa/Asir, 53 seconds on Faug\`ere FGb
 \item Current  \item Current
   
 $F_4$ in FGb : much more efficient than $F_4$ in Risa/Asir  $F_4$ in FGb : much more efficient than $F_4$ in Risa/Asir
   
 Buchberger in Singular [Singular] : faster than Risa/Asir  Buchberger in Singular ({\tt http://www.singular.uni-kl.de})
   : faster than Risa/Asir
   
 $\Leftarrow$ efficient monomial and polynomial comutation  
   
   $\Leftarrow$ efficient monomial and polynomial computation
   
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Some timing data --- DRL Groebner basis computation}  \fbox{Timing data --- DRL Groebner basis computation}
   
 \underline{Over $GF(32003)$}  \underline{Over $GF(32003)$}
 \begin{center}  \begin{center}
Line 263  $\Leftarrow$ efficient monomial and polynomial comutat
Line 320  $\Leftarrow$ efficient monomial and polynomial comutat
                 & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline                  & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
 Asir $Buchberger$       & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline  Asir $Buchberger$       & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline
 Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline  Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
 CoCoA 4 & 241 & & 3.8 & 35 & 402 & &  \\ \hline\hline  CoCoA 4 & 241 & & 3.8 & 35 & 402 & & --- \\ \hline\hline
 Asir $F_4$      & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline  Asir $F_4$      & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline
 FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline  FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
 \end{tabular}  \end{tabular}
Line 275  FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \
Line 332  FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|} \hline
                 & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline                  & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
 Asir $Buchberger$       & 389 & 594 & 29 & 299 & 34950 \\ \hline  Asir $Buchberger$       & 389 & 594 & 29 & 299 & 34950 \\ \hline
 Singular & & 15247 & 7.6 & 79 & \\ \hline  Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
 CoCoA 4 & & & 57 & 709 & \\ \hline\hline  CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
 Asir $F_4$      &  989 & 456 & 90 & 991 & 4939 \\ \hline  Asir $F_4$      &  989 & 456 & 90 & 991 & 4939 \\ \hline
 FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline  FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   --- : not tested
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{How do we proceed?}  \fbox{How do we proceed?}
   
   \underline{Total performance : not excellent, but not so bad}
   
 \begin{itemize}  \begin{itemize}
   \item Trying to improve our implementation
   
   This is very important as a motivation of further development
   
   \begin{itemize}
   
   \item Computation of $b$-function
   
   fast but not satisfactory
   
   $\Rightarrow$ Groebner basis computation in Weyl
   algebra should be improved
   \end{itemize}
   
 \item Developing new OpenXM servers  \item Developing new OpenXM servers
   
 {ox\_NTL} for univariate factorization,  {ox\_NTL} for univariate factorization,
Line 295  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
Line 368  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
   
 $\Rightarrow$ Risa/Asir can be a front-end of efficient servers  $\Rightarrow$ Risa/Asir can be a front-end of efficient servers
   
 \item Trying to improve our implementation  
   
 This is very important as a motivation of further development  
   
 Computation of $b$-function : still faster than any other system  
 (Kan/sm1, Macaulay2) but not satisfactory  
   
 $\Rightarrow$ Groebner basis computation in Weyl  
 algebra should be improved  
 \end{itemize}  \end{itemize}
   
 \begin{center}  \begin{center}
Line 358  algebra should be improved
Line 422  algebra should be improved
 %\begin{itemize}  %\begin{itemize}
 %\item Stack = I/O buffer for (possibly large) objects  %\item Stack = I/O buffer for (possibly large) objects
 %  %
 %Multiple requests can be sent before their exection  %Multiple requests can be sent before their execution
 %  %
 %A server does not get stuck in sending results  %A server does not get stuck in sending results
 %\end{itemize}  %\end{itemize}
Line 393  def grvsf4(G,V,M,O)
Line 457  def grvsf4(G,V,M,O)
 \begin{slide}{}  \begin{slide}{}
 \fbox{References}  \fbox{References}
   
 [Bernardin] L. Bernardin, On square-free factorization of  [BERN97] L. Bernardin, On square-free factorization of
 multivariate polynomials over a finite field, Theoretical  multivariate polynomials over a finite field, Theoretical
 Computer Science 187 (1997), 105-116.  Computer Science 187 (1997), 105-116.
   
 [Boehm] {\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc}  [FAUG99] J.C. Faug\`ere,
   
 [Faug\`ere] J.C. Faug\`ere,  
 A new efficient algorithm for computing Groebner bases  ($F_4$),  A new efficient algorithm for computing Groebner bases  ($F_4$),
 Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.  Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.
   
 [Hoeij] M. van Heoij, Factoring polynomials and the knapsack problem,  [GRAY98] S. Gray et al,
   Design and Implementation of MP, A Protocol for Efficient Exchange of
   Mathematical Expression,
   J. Symb. Comp. {\bf 25} (1998), 213-238.
   
   [HOEI00] 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).
   
 [Noro] M. Noro, J. McKay,  [LIAO01] W. Liao et al,
   OMEI: An Open Mathematical Engine Interface,
   Proc. ASCM2001 (2001), 82-91.
   [NORO97] M. Noro, J. McKay,
 Computation of replicable functions on Risa/Asir.  Computation of replicable functions on Risa/Asir.
 Proc. of PASCO'97, ACM Press, 130-138 (1997).  Proc. PASCO'97, ACM Press (1997), 130-138.
   \end{slide}
   
 [NY] M. Noro, K. Yokoyama,  \begin{slide}{}
   
   [NOYO99] 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.
 \end{slide}  
   
 \begin{slide}{}  [OAKU97] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
   
 [Oaku] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic  
 local cohomology groups of $D$-modules.  local cohomology groups of $D$-modules.
 Advancees in Applied Mathematics, 19 (1997), 61-105.  Advances in Applied Mathematics, 19 (1997), 61-105.
   
 [OpenMath] {\tt http://www.openmath.org}  [ROUI96] F. Rouillier,
   
 [OpenXM] {\tt http://www.openxm.org}  
   
 [PARI] {\tt http://www.parigp-home.de}  
   
 [Risa/Asir] {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}  
   
 [Rouillier] F. Rouillier,  
 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.  [SHYO96] 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}  [TRAV88] 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.  [WANG99] P. S. Wang,
   Design and Protocol for Internet Accessible Mathematical Computation,
   Proc. ISSAC '99 (1999), 291-298.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
Line 491  Berlekamp-Zassenhaus
Line 554  Berlekamp-Zassenhaus
   
 Trager's algorithm + some improvement  Trager's algorithm + some improvement
   
 \item Over finite fieds  \item Over finite fields
   
 DDF + Cantor-Zassenhaus; FFT for large finite fields  DDF + Cantor-Zassenhaus; FFT for large finite fields
 \end{itemize}  \end{itemize}
Line 503  DDF + Cantor-Zassenhaus; FFT for large finite fields
Line 566  DDF + Cantor-Zassenhaus; FFT for large finite fields
   
 Classical EZ algorithm  Classical EZ algorithm
   
 \item Over small finite fieds  \item Over small finite fields
   
 Modified Bernardin's square free algorithm [Bernardin],  Modified Bernardin's square free algorithm [BERN97],
   
 possibly Hensel lifting over extension fields  possibly Hensel lifting over extension fields
 \end{itemize}  \end{itemize}
Line 548  Key : an efficient implementation of Leibniz rule
Line 611  Key : an efficient implementation of Leibniz rule
 \begin{itemize}  \begin{itemize}
 \item More efficient than our Buchberger algorithm implementation  \item More efficient than our Buchberger algorithm implementation
   
 but less efficient than FGb by Faugere  but less efficient than FGb by Faug\`ere
 \end{itemize}  \end{itemize}
   
 \item Over the rationals  \item Over the rationals
Line 689  Writes to the descriptor 4
Line 752  Writes to the descriptor 4
   
 In Risa/Asir subroutine library {\tt libasir.a}:  In Risa/Asir subroutine library {\tt libasir.a}:
   
 OpenXM functionalities are implemented as functon calls  OpenXM functionalities are implemented as function calls
   
 pushing and popping data, executing stack commands etc.  pushing and popping data, executing stack commands etc.
 \end{itemize}  \end{itemize}
Line 724  Competitive computation is easily realized ($\Rightarr
Line 787  Competitive computation is easily realized ($\Rightarr
   
 \begin{enumerate}  \begin{enumerate}
 \item (C $\rightarrow$ S) Arguments are sent in binary encoded form.  \item (C $\rightarrow$ S) Arguments are sent in binary encoded form.
 \item (C $\rightarrow$ S) The number of aruments is sent as {\sl Integer32}.  \item (C $\rightarrow$ S) The number of arguments is sent as {\sl Integer32}.
 \item (C $\rightarrow$ S) A function name is sent as {\sl Cstring}.  \item (C $\rightarrow$ S) A function name is sent as {\sl Cstring}.
 \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.  \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.
 \item The result is pushed to the stack.  \item The result is pushed to the stack.
Line 740  conversion are necessary.
Line 803  conversion are necessary.
 \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}  \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}
   
 \begin{enumerate}  \begin{enumerate}
 \item (C $\rightarrow$ S) A character string represeting a request in a server's  \item (C $\rightarrow$ S) A character string representing a request in a server's
 user language is sent as {\sl Cstring}.  user language is sent as {\sl Cstring}.
 \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.  \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.
 \item The result is pushed to the stack.  \item The result is pushed to the stack.
Line 763  enough to read the result.
Line 826  enough to read the result.
 %\item 1989--1992  %\item 1989--1992
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC [Boehm]  %\item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC
 %  %
 %\item Developed univariate and multivariate factorizers over the rationals.  %\item Developed univariate and multivariate factorizers over the rationals.
 %\end{itemize}  %\end{itemize}
Line 775  enough to read the result.
Line 838  enough to read the result.
 %  %
 %Written in user language $\Rightarrow$ rewritten in C (by Murao)  %Written in user language $\Rightarrow$ rewritten in C (by Murao)
 %  %
 %$\Rightarrow$ trace lifting [Traverso]  %$\Rightarrow$ trace lifting [TRAV88]
 %  %
 %\item Univariate factorization over algebraic number fields  %\item Univariate factorization over algebraic number fields
 %  %
Line 794  enough to read the result.
Line 857  enough to read the result.
 %\item Primary ideal decomposition  %\item Primary ideal decomposition
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Shimoyama-Yokoyama algorithm [SY]  %\item Shimoyama-Yokoyama algorithm [SHYO96]
 %\end{itemize}  %\end{itemize}
 %  %
 %\item Improvement of Buchberger algorithm  %\item Improvement of Buchberger algorithm
Line 806  enough to read the result.
Line 869  enough to read the result.
 %  %
 %\item Modular change of ordering, Modular RUR  %\item Modular change of ordering, Modular RUR
 %  %
 %These are joint works with Yokoyama [NY]  %These are joint works with Yokoyama [NOYO99]
 %\end{itemize}  %\end{itemize}
 %\end{itemize}  %\end{itemize}
 %  %
Line 816  enough to read the result.
Line 879  enough to read the result.
 %\fbox{History of development : 1996-1998}  %\fbox{History of development : 1996-1998}
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Distributed compuatation  %\item Distributed computation
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item A prototype of OpenXM  %\item A prototype of OpenXM
Line 825  enough to read the result.
Line 888  enough to read the result.
 %\item Improvement of Buchberger algorithm  %\item Improvement of Buchberger algorithm
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Content reduction during nomal form computation  %\item Content reduction during normal form computation
 %  %
 %\item Its parallelization by the above facility  %\item Its parallelization by the above facility
 %  %
 %\item Computation of odd order replicable functions [Noro]  %\item Computation of odd order replicable functions [NORO97]
 %  %
 %Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})  %Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})
 %  %
Line 858  enough to read the result.
Line 921  enough to read the result.
 %\begin{itemize}  %\begin{itemize}
 %\item OpenXM specification was written by Noro and Takayama  %\item OpenXM specification was written by Noro and Takayama
 %  %
 %Borrowed idea on encoding, phrase book from OpenMath [OpenMath]  %Borrowed idea on encoding, phrase book from OpenMath
 %  %
 %\item Functions for distributed computation were rewritten  %\item Functions for distributed computation were rewritten
 %\end{itemize}  %\end{itemize}
Line 874  enough to read the result.
Line 937  enough to read the result.
 %\item Test implementation of $F_4$  %\item Test implementation of $F_4$
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Implemented according to [Faug\`ere]  %\item Implemented according to [FAUG99]
 %  %
 %\item Over $GF(p)$ : pretty good  %\item Over $GF(p)$ : pretty good
 %  %
Line 891  enough to read the result.
Line 954  enough to read the result.
 %\begin{itemize}  %\begin{itemize}
 %\item Noro moved from Fujitsu to Kobe university  %\item Noro moved from Fujitsu to Kobe university
 %  %
 %Started Kobe branch [Risa/Asir]  %Started Kobe branch
 %\end{itemize}  %\end{itemize}
 %  %
 %\item OpenXM [OpenXM]  %\item OpenXM
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Revising the specification : OX-RFC100, 101, (102)  %\item Revising the specification : OX-RFC100, 101, (102)
Line 905  enough to read the result.
Line 968  enough to read the result.
 %\item Weyl algebra  %\item Weyl algebra
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Buchberger algorithm [Takayama]  %\item Buchberger algorithm [TAKA90]
 %  %
 %\item $b$-function computation [Oaku]  %\item $b$-function computation [OAKU97]
 %  %
 %Minimal polynomial computation by modular method  %Minimal polynomial computation by modular method
 %\end{itemize}  %\end{itemize}
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  Added in v.1.7
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