=================================================================== RCS file: /home/cvs/OpenXM/doc/Papers/Attic/dagb-noro.tex,v retrieving revision 1.5 retrieving revision 1.9 diff -u -p -r1.5 -r1.9 --- OpenXM/doc/Papers/Attic/dagb-noro.tex 2001/10/09 01:44:21 1.5 +++ OpenXM/doc/Papers/Attic/dagb-noro.tex 2001/10/11 08:43:08 1.9 @@ -1,4 +1,4 @@ -% $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.4 2001/10/04 08:22:20 noro Exp $ +% $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.8 2001/10/11 01:34:42 noro Exp $ \setlength{\parskip}{10pt} \begin{slide}{} @@ -7,70 +7,46 @@ \end{center} \end{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{A computer algebra system Risa/Asir} -\begin{itemize} -\item An environment for parallel distributed computation +({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}) -Both for interactive, non-interactive environment - -\item Client-server architecture - -Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server - -OX (OpenXM) message : command and data - -\item Data - -Encoding : CMO (Common Mathematical Object format) - -Serialized representation of mathematical object - ---- Main idea was borrowed from OpenMath [OpenMath] -\item Command - -stack machine command --- server is a stackmachine - -+ server's own command sequences --- hybrid server -\end{itemize} -\end{slide} - - -\begin{slide}{} -\fbox{OpenXM and OpenMath} - \begin{itemize} -\item OpenMath +\item Software mainly for polynomial computation -\begin{itemize} -\item A standard for representing mathematical objects - -\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 - \item User language with C-like syntax C language without type declaration, with list processing @@ -81,73 +57,86 @@ C language without type declaration, with list process Whole source tree is available via CVS +The latest version : see {\tt http://www.openxm.org} + \item OpenXM interface \begin{itemize} +\item OpenXM + +An infrastructure for exchanging mathematical data \item Risa/Asir is a main client in OpenXM package. \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{slide} \begin{slide}{} -\fbox{Aim of developing Risa/Asir} +\fbox{Goal of developing Risa/Asir} \begin{itemize} -\item Efficient implementation in specific area +\item Testing new algorithms -Polynomial factorization, Groebner basis related computation - -$\Rightarrow$ serves as an OpenXM server/library - -\item Front-end of a general purpose math software - -Risa/Asir contains PARI library [PARI] from the very beginning - -It also acts as a main client of OpenXM package - -\end{itemize} -\end{slide} - -\begin{slide}{} -\fbox{Capability for polynomial computation} - \begin{itemize} -\item Fundamental polynomial arithmetics +\item Development started in Fujitsu labs -recursive representaion and distributed representation - -\item Polynomial factorization - -\begin{itemize} -\item Univariate : over {\bf Q}, algebraic number fields and finite fields - -\item Multivariate : over {\bf Q} +Polynomial factorization, Groebner basis related computation, +cryptosystems , quantifier elimination , $\ldots$ \end{itemize} -\item Groebner basis computation +\item To be a general purpose, open system -\begin{itemize} -\item Buchberger and $F_4$ [Faug\'ere] algorithm +Since 1997, we have been developing OpenXM package +containing various servers and clients -\item Change of ordering/RUR [Rouillier] of 0-dimensional ideals +Risa/Asir is a component of OpenXM -\item Primary ideal decomposition +\item Environment for parallel and distributed computation -\item Computation of $b$-function (in Weyl Algebra) \end{itemize} -\end{itemize} \end{slide} +%\begin{slide}{} +%\fbox{Capability for polynomial computation} +% +%\begin{itemize} +%\item Fundamental polynomial arithmetics +% +%recursive representation and distributed representation +% +%\item Polynomial factorization +% +%\begin{itemize} +%\item Univariate : over {\bf Q}, algebraic number fields and finite fields +% +%\item Multivariate : over {\bf Q} +%\end{itemize} +% +%\item Groebner basis computation +% +%\begin{itemize} +%\item Buchberger and $F_4$ [FAUG99] algorithm +% +%\item Change of ordering/RUR [ROUI96] of 0-dimensional ideals +% +%\item Primary ideal decomposition +% +%\item Computation of $b$-function (in Weyl Algebra) +%\end{itemize} +%\end{itemize} +%\end{slide} + \begin{slide}{} \fbox{History of development : Polynomial factorization} \begin{itemize} \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 Univariate and multivariate factorizers over {\bf Q} @@ -162,6 +151,8 @@ Intensive use of successive extension, non-squarefree Univariate factorization over large finite fields +Motivated by a reseach project in Fujitsu on cryptography + \item 2000-current Multivariate factorization over small finite fields (in progress) @@ -174,165 +165,145 @@ Multivariate factorization over small finite fields (i \begin{itemize} \item 1992-1994 -User language $\Rightarrow$ C version; trace lifting [Traverso] +User language $\Rightarrow$ C version; trace lifting [TRAV88] \item 1994-1996 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[ROUI96] [NOYO99] -Primary ideal decompositon [SY] +Primary ideal decomposition [SHYO96] \item 1996-1998 -Effifcient content reduction during NF computation and its parallelization -[Noro] (Solved {\it McKay} system for the first time) +Efficient content reduction during NF computation [NORO97] +Solved {\it McKay} system for the first time \item 1998-2000 -Test implementation of $F_4$ +Test implementation of $F_4$ [FAUG99] \item 2000-current -Buchberger algorithm in Weyl algebra [Takayama] +Buchberger algorithm in Weyl algebra -Efficient $b$-function computation by a modular method +Efficient $b$-function computation[OAKU97] by a modular method \end{itemize} \end{slide} \begin{slide}{} -\fbox{Performance --- Factorizer} +\fbox{Timing data --- Factorization} -\begin{itemize} -\item 4 years ago +\underline{Univariate; over {\bf Q}} -Over {\bf Q} : fine compared with existing software -like REDUCE, Mathematica, maple +$N_i$ : a norm of a polynomial, $\deg(N_i) = i$ +\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} -Univarate, over algebraic number fields : -fine because of some tricks for polynomials -derived from norms. +\underline{Multivariate; over {\bf Q}} -\item Current +$W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ 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 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline +Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline +\end{tabular} +\end{center} -Multivariate : not so bad - -Univariate : completely obsolete by M. van Hoeij's new algorithm -[Hoeij] -\end{itemize} - +%--- : not tested \end{slide} \begin{slide}{} -\fbox{Performance --- Groebner basis related computation} +\fbox{Timing data --- DRL Groebner basis computation} -\begin{itemize} -\item 7 years ago +\underline{Over $GF(32003)$} +\begin{center} +\begin{tabular}{|c||c|c|c|c|c|c|c|} \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 +Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline +CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\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 +\end{tabular} +\end{center} -Trace lifting : rather fine but coefficient swells often occur +\underline{Over {\bf Q}} -Homogenization+trace lifting : robust and fast in the above cases - -\item 4 years ago - -Modular RUR was comparable with Rouillier's implementation. - -DRL basis of {\it McKay}: - -5 days on Risa/Asir, 53 seconds on Faugere FGb -\item Current - -$F_4$ in FGb : much more efficient than $F_4$ in Risa/Asir - -Buchberger in Singular [Singular] : faster than Risa/Asir - -$\Leftarrow$ efficient monomial and polynomial representation - -\end{itemize} -\end{slide} - -\begin{slide}{} -\fbox{How do we proceed?} - -\begin{itemize} -\item Developing new OpenXM servers - -{ox\_NTL} for univariate factorization, - -{ox\_FGb} for Groebner basis computation (is it possible?) etc. - -$\Rightarrow$ Risa/Asir can be a front-end of efficient servers - -\item Trying to improve our implementation - -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} - \begin{center} -\underline{In both cases, OpenXM interface is important} +\begin{tabular}{|c||c|c|c|c|c|} \hline + & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline +Asir $Buchberger$ & 389 & 594 & 29 & 299 & 34950 \\ \hline +Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline +CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline +Asir $F_4$ & 989 & 456 & 90 & 991 & 4939 \\ \hline +FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline +\end{tabular} \end{center} +--- : not tested \end{slide} \begin{slide}{} -\fbox{OpenXM server interface in Risa/Asir} +\fbox{Summary of performance} \begin{itemize} -\item TCP/IP stream +\item Factorizer \begin{itemize} -\item Launcher +\item Multivariate : reasonable performance -A client executes a launcher on a host. +\item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00] +\end{itemize} -The launcher launches a server on the same host. +\item Groebner basis computation -\item Server +\begin{itemize} +\item Buchberger -Reads from the descriptor 3 +Singular shows nice perfomance -Writes to the descriptor 4 +Trace lifting is efficient in some cases over {\bf Q} -\end{itemize} +\item $F_4$ -\item Subroutine call +FGb is much faster than Risa/Asir -In Risa/Asir subroutine library {\tt libasir.a}: - -OpenXM functionalities are implemented as functon calls - -pushing and popping data, executing stack commands etc. +But we observe that {\it McKay} is computed efficiently by $F_4$ \end{itemize} +\end{itemize} + \end{slide} \begin{slide}{} -\fbox{OpenXM client interface in Risa/Asir} +\fbox{Summary} \begin{itemize} -\item Primitive interface functions +\item Total performance is not excellent, but not so bad -Pushing and popping data, sending commands etc. +\item A completely open system -\item Convenient functions +The whole source is available -Launching servers, +\item Interface compliant to OpenXM RFC-100 -Calling remote functions, - -Resetting remote executions etc. - -\item Parallel distributed computation - -Simple parallelization is practically important - -Competitive computation is easily realized ($\Rightarrow$ demo) +The interface is fully documented \end{itemize} + \end{slide} @@ -382,7 +353,7 @@ Competitive computation is easily realized ($\Rightarr %\begin{itemize} %\item Stack = I/O buffer for (possibly large) objects % -%Multiple requests can be sent before their exection +%Multiple requests can be sent before their execution % %A server does not get stuck in sending results %\end{itemize} @@ -390,6 +361,38 @@ Competitive computation is easily realized ($\Rightarr %\end{slide} \begin{slide}{} +\fbox{OpenXM (Open message eXchange protocol for Mathematics) } + +\begin{itemize} +\item An environment for parallel distributed computation + +Both for interactive, non-interactive environment + +\item OpenXM RFC-100 = Client-server architecture + +Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server + +OX (OpenXM) message : command and data + +\item Data + +Encoding : CMO (Common Mathematical Object format) + +Serialized representation of mathematical object + +--- Main idea was borrowed from OpenMath + +({\tt http://www.openmath.org}) + +\item Command + +stack machine command --- server is a stackmachine + ++ server's own command sequences --- hybrid server +\end{itemize} +\end{slide} + +\begin{slide}{} \fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ } \begin{verbatim} @@ -417,53 +420,52 @@ def grvsf4(G,V,M,O) \begin{slide}{} \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 Computer Science 187 (1997), 105-116. -[Boehm] {\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc} - -[Faug\`ere] J.C. Faug\`ere, +[FAUG99] J.C. Faug\`ere, A new efficient algorithm for computing Groebner bases ($F_4$), 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 Hoeij, Factoring polynomials and the knapsack problem, 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. -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 Representation of Zero-Dimensional Ideals. J. Symb. Comp. {\bf 28}/1 (1999), 243-263. -\end{slide} -\begin{slide}{} - -[Oaku] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic +[OAKU97] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic 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} - -[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, +[ROUI96] F. Rouillier, R\'esolution des syst\`emes z\'ero-dimensionnels. 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} \begin{slide}{} @@ -515,7 +517,7 @@ Berlekamp-Zassenhaus Trager's algorithm + some improvement -\item Over finite fieds +\item Over finite fields DDF + Cantor-Zassenhaus; FFT for large finite fields \end{itemize} @@ -527,9 +529,9 @@ DDF + Cantor-Zassenhaus; FFT for large finite fields 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 \end{itemize} @@ -572,7 +574,7 @@ Key : an efficient implementation of Leibniz rule \begin{itemize} \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} \item Over the rationals @@ -689,11 +691,66 @@ The knapsack factorization is available via {\tt pari( \end{slide} \begin{slide}{} +\fbox{OpenXM server interface in Risa/Asir} + +\begin{itemize} +\item TCP/IP stream + +\begin{itemize} +\item Launcher + +A client executes a launcher on a host. + +The launcher launches a server on the same host. + +\item Server + +Reads from the descriptor 3 + +Writes to the descriptor 4 + +\end{itemize} + +\item Subroutine call + +In Risa/Asir subroutine library {\tt libasir.a}: + +OpenXM functionalities are implemented as function calls + +pushing and popping data, executing stack commands etc. +\end{itemize} +\end{slide} + +\begin{slide}{} +\fbox{OpenXM client interface in Risa/Asir} + +\begin{itemize} +\item Primitive interface functions + +Pushing and popping data, sending commands etc. + +\item Convenient functions + +Launching servers, + +Calling remote functions, + +Resetting remote executions etc. + +\item Parallel distributed computation + +Simple parallelization is practically important + +Competitive computation is easily realized ($\Rightarrow$ demo) +\end{itemize} +\end{slide} + +\begin{slide}{} \fbox{Executing functions on a server (I) --- {\tt SM\_executeFunction}} \begin{enumerate} \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 command {\tt SM\_executeFunction} is sent. \item The result is pushed to the stack. @@ -709,7 +766,7 @@ conversion are necessary. \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}} \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}. \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent. \item The result is pushed to the stack. @@ -732,7 +789,7 @@ enough to read the result. %\item 1989--1992 % %\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. %\end{itemize} @@ -744,7 +801,7 @@ enough to read the result. % %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 % @@ -763,7 +820,7 @@ enough to read the result. %\item Primary ideal decomposition % %\begin{itemize} -%\item Shimoyama-Yokoyama algorithm [SY] +%\item Shimoyama-Yokoyama algorithm [SHYO96] %\end{itemize} % %\item Improvement of Buchberger algorithm @@ -775,7 +832,7 @@ enough to read the result. % %\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} % @@ -785,7 +842,7 @@ enough to read the result. %\fbox{History of development : 1996-1998} % %\begin{itemize} -%\item Distributed compuatation +%\item Distributed computation % %\begin{itemize} %\item A prototype of OpenXM @@ -794,11 +851,11 @@ enough to read the result. %\item Improvement of Buchberger algorithm % %\begin{itemize} -%\item Content reduction during nomal form computation +%\item Content reduction during normal form computation % %\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}) % @@ -827,7 +884,7 @@ enough to read the result. %\begin{itemize} %\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 %\end{itemize} @@ -843,7 +900,7 @@ enough to read the result. %\item Test implementation of $F_4$ % %\begin{itemize} -%\item Implemented according to [Faug\`ere] +%\item Implemented according to [FAUG99] % %\item Over $GF(p)$ : pretty good % @@ -860,10 +917,10 @@ enough to read the result. %\begin{itemize} %\item Noro moved from Fujitsu to Kobe university % -%Started Kobe branch [Risa/Asir] +%Started Kobe branch %\end{itemize} % -%\item OpenXM [OpenXM] +%\item OpenXM % %\begin{itemize} %\item Revising the specification : OX-RFC100, 101, (102) @@ -874,9 +931,9 @@ enough to read the result. %\item Weyl algebra % %\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 %\end{itemize}