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% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.9 2001/12/28 06:06:15 noro Exp $

% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.10 2002/01/04 06:06:09 noro Exp $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This is a sample input file for your contribution to a multi-

% author book to be published by Springer Verlag.

Line 60

\usepackage{epsfig}

\def\cont{{\rm cont}}

\def\GCD{{\rm GCD}}

\def\Q{{\bf Q}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 87

Line 88

\maketitle % typesets the title of the contribution

\begin{abstract}

%\begin{abstract}

Risa/Asir is software for polynomial computation. It has been

%Risa/Asir is software for polynomial computation. It has been

developed for testing experimental polynomial algorithms, and now it

%developed for testing experimental polynomial algorithms, and now it

acts also as a main component in the OpenXM package \cite{OPENXM}.

%acts also as a main component in the OpenXM package \cite{noro:OPENXM}.

OpenXM is an infrastructure for exchanging mathematical

%OpenXM is an infrastructure for exchanging mathematical

data. It defines a client-server architecture for parallel and

%data. It defines a client-server architecture for parallel and

distributed computation. In this article we present an overview of

%distributed computation. In this article we present an overview of

Risa/Asir and review several techniques for improving performances of

%Risa/Asir and review several techniques for improving performances of

Groebner basis computation over {\bf Q}. We also show Risa/Asir's

%Groebner basis computation over {\bf Q}. We also show Risa/Asir's

OpenXM interfaces and their usages.

%OpenXM interfaces and their usages.

\end{abstract}

%\end{abstract}

\section{A computer algebra system Risa/Asir}

\section{Introduction}

\subsection{What is Risa/Asir?}

%Risa/Asir $B$O(B, $B?t(B, $BB?9`<0$J$I$KBP$9$k1i;;$r<BAu$9$k(B engine,

%$B%f!<%68@8l$r<BAu$9$k(B parser and interpreter $B$*$h$S(B,

%$BB>$N(B application $B$H$N(B interaction $B$N$?$a$N(B OpenXM interface $B$+$i$J$k(B

%computer algebra system $B$G$"$k(B.

Risa/Asir is a computer algebra system which consists of an engine for

operations on numbers and polynomials, a parser and an interpreter for

the user language, and a communication interface called OpenXM API for

interaction with other applications.

%engine $B$G$O(B, $B?t(B, $BB?9`<0$J$I$N(B arithmetics $B$*$h$S(B, $BB?9`<0(B

%GCD, $B0x?tJ,2r(B, $B%0%l%V%J4pDl7W;;$,<BAu$5$l$F$$$k(B. $B$3$l$i$OAH$_9~$_4X?t(B

%$B$H$7$F%f!<%68@8l$+$i8F$S=P$5$l$k(B.

The engine implements fundamental arithmetics on numbers and polynomials,

polynomial GCD, polynomial factorizations and Groebner basis computations,

etc. These can be called from the user language as builtin functions.

%Risa/Asir $B$N%f!<%68@8l$O(B C $B8@8l(B like $B$JJ8K!$r$b$A(B, $BJQ?t$N7?@k8@$,(B

%$B$J$$(B, $B%j%9%H=hM}$*$h$S<+F0(B garbage collection $B$D$-$N%$%s%?%W%j%?(B

%$B8@8l$G$"$k(B. $B%f!<%68@8l%W%m%0%i%`$O(B parser $B$K$h$jCf4V8@8l$K(B

%$BJQ49$5$l(B, interpreter $B$K$h$j2r<a<B9T$5$l$k(B. interpreter $B$K$O(B

%gdb $BIw$N(B debugger $B$,AH$_9~$^$l$F$$$k(B.

The user language has C-like syntax, without type declarations

of variables, with list processing and with automatic garbage collection.

The interpreter is equipped with a {\tt gdb}-like debugger.

%$B$3$l$i$N5!G=$O(B OpenXM interface $B$rDL$7$FB>$N(B application $B$+$i$b;HMQ2D(B

%$BG=$G$"$k(B. OpenXM \cite{noro:RFC100} $B$O?t3X%=%U%H%&%'%"$N(B client-server

%$B7?$NAj8_8F$S=P$7$N$?$a$N(B $B%W%m%H%3%k$G$"$k(B.

These functions can be called from other applications via OpenXM API.

OpenXM \cite{noro:RFC100} is a protocol for client-server

communications between mathematical software. We are distributing

OpenXM package \cite{noro:OPENXM}, which is a collection of various

clients and servers comlient to the OpenXM protocol specification.

Risa/Asir \cite{RISA} is software mainly for polynomial

%Risa/Asir $B$OB?9`<00x?tJ,2r(B, $B%,%m%"727W;;(B \cite{noro:ANY}, $B%0%l%V%J4pDl(B

computation. Its major functions are polynomial factorization and

%$B7W;;(B \cite{noro:NM,noro:NY}, $B=`AG%$%G%"%kJ,2r(B \cite{noro:SY}, $B0E9f(B

Groebner basis computation, whose core parts are implemented as

%\cite{noro:IKNY} $B$K$*$1$k<B83E*%"%k%4%j%:%`(B $B$r%F%9%H$9$k$?$a$N%W%i%C%H(B

built-in functions. Some higher algorithms such as primary ideal

%$B%U%)!<%`$H$7$F3+H/$5$l$F$-$?(B. $B$^$?(B, OpenXM API $B$rMQ$$$F(B parallel

decomposition or Galois group computation are built on them by the

%distributed computation $B$N<B83$K$bMQ$$$i$l$F$$$k(B. $B:,44$r$J$9$N$OB?9`(B

user language called Asir language. Asir language can be regarded as C

%$B<00x?tJ,2r$*$h$S%0%l%V%J4pDl7W;;$G$"$k(B. $BK\9F$G$O(B, $BFC$K(B, $B%0%l%V%J4pDl(B

language without type declaration of variables, with list processing,

%$B7W;;$K4X$7$F(B, $B$=$N4pK\$*$h$S(B {\bf Q} $B>e$G$N7W;;$N:$Fq$r9nI~$9$k$?$a$N(B

and with automatic garbage collection. A built-in {\tt gdb}-like user

%$B$5$^$6$^$J9)IW$*$h$S$=$N8z2L$K$D$$$F=R$Y$k(B. $B$^$?(B, Risa/Asir $B$O(B OpenXM

language debugger is available. Risa/Asir is open source and the

%package $B$K$*$1$k<gMW$J(B component $B$N0l$D$G$"$k(B. Risa/Asir $B$r(B client $B$"(B

source code and binaries are available via {\tt ftp} or {\tt CVS}.

%$B$k$$$O(B server $B$H$7$FMQ$$$kJ,;6JBNs7W;;$K$D$$$F(B, $B<BNc$r$b$H$K2r@b$9$k(B.

Risa/Asir is not only a standalone computer algebra system but also a

Risa/Asir has been used for implementing and testing experimental

main component in OpenXM package \cite{OPENXM}, which is a collection

algorithms such as polynomial factorizations, splitting field and

of various software compliant to OpenXM protocol specification.

Galois group computations \cite{noro:ANY}, Groebner basis computations

OpenXM is an infrastructure for exchanging mathematical data and

\cite{noro:REPL,noro:NOYO} primary ideal decomposition \cite{noro:SY}

Risa/Asir has three kinds of OpenXM interfaces :

and cryptgraphy \cite{noro:IKNY}. In these applications the important

OpenXM API in the Risa/Asir user language,

funtions are polynomial factorization and Groebner basis

OpenXM C language API in the Risa/Asir subroutine library,

computation. We focus on Groebner basis computation and we review its

and an OpenXM server.

fundamentals and vaious efforts for improving efficiency especially

Our goals of developing Risa/Asir are as follows:

over $\Q$. Risa/Asir is also a main component of OpenXM package and

it has been used for parallel distributed computation with OpenXM API.

We will explain how one can execute parallel distributed computation

by using Risa/Asir as a client or a server.

\begin{enumerate}

\section{Efficient Groebner basis computation over {\bf Q}}

\item Providing a platform for testing new algorithms

\label{tab:gbtech}

Risa/Asir has been a platform for testing experimental algorithms in

polynomial factorization, Groebner basis computation,

cryptography and quantifier elimination. As to Groebner basis, we have

been mainly interested in problems over {\bf Q} and we tried applying

various modular techniques to overcome difficulties caused by huge

intermediate coefficients. We have had several results and they have

been implemented in Risa/Asir with other known methods.

\item General purpose open system

We need a lot of functions to make Risa/Asir a general purpose

computer algebra system. In recent years we can make use of various high

performance applications or libraries as free software. We wrapped

such software as OpenXM servers and we started to release a collection

of such servers and clients as the OpenXM package in 1997. Risa/Asir

is now a main client in the package.

\item Environment for parallel and distributed computation

The ancestor of OpenXM is a protocol designed for doing parallel

distributed computations by connecting multiple Risa/Asir's over

TCP/IP. OpenXM is also designed to provide an environment for

efficient parallel distributed computation. Currently only

client-server communication is available, but we are preparing a

specification OpenXM-RFC 102 allowing client-client communication,

which will enable us to execute wider range of parallel algorithms

requiring collective operations efficiently.

\end{enumerate}

\subsection{Groebner basis and the related computation}

Currently Risa/Asir can only deal with polynomial ring. Operations on

modules over polynomial rings have not yet supported. However, both

commutative polynomial rings and Weyl algebra are supported and one

can compute Groebner basis in both rings over {\bf Q}, fields of

rational functions and finite fields. In the early stage of our

development, our effort was mainly devoted to improve the efficiency

of computation over {\bf Q}. Our main tool is modular

computation. For Buchberger algorithm we adopted the trace lifting

algorithm by Traverso \cite{TRAV} and elaborated it by applying our

theory on a correspondence between Groebner basis and its modular

image \cite{NOYO}. We also combine the trace lifting with

homogenization to stabilize selection strategies, which enables us to

compute several examples efficiently which are hard to compute without

such a combination. Our modular method can be applied to the change

of ordering algorithm\cite{FGLM} and rational univariate

representation \cite{RUR}. We also made a test implementation of

$F_4$ algorithm \cite{F4}. In the later section we will show timing

data on Groebner basis computation.

\subsection{Polynomial factorization}

Here we briefly review functions on polynomial factorization. For

univariate factorization over {\bf Q}, the Berlekamp-Zassenhaus

algorithm is implemented. Efficient algorithms recently proposed have

not yet implemented. For univariate factorization over algebraic

number fields, Trager's algorithm \cite{TRAGER} is implemented with

some modifications. Its major applications are splitting field and

Galois group computation of polynomials over {\bf Q} \cite{ANY}. For

such purpose a tower of simple extensions are suitable because factors

represented over a simple extension often have huge coefficients. For

univariate factorization over finite fields, equal degree

factorization and Cantor-Zassenhaus algorithm are implemented. We can

use various representation of finite fields: $GF(p)$ with a machine

integer prime $p$, $GF(p)$ and $GF(p^n)$ with any odd prime $p$,

$GF(2^n)$ with a bit-array representation of polynomials over $GF(2)$

and $GF(p^n)$ with small $p^n$ represented by a primitive root. For

multivariate factorization over {\bf Q}, the EZ(Extended Zassenhaus)

type algorithm is implemented.

\subsection{Other functions}

By applying Groebner basis computation and polynomial factorization,

we have implemented several higher level algorithms. A typical

application is primary ideal decomposition of polynomial ideals over

{\bf Q}, which needs both functions. Shimoyama-Yokoyama algorithm

\cite{SY} for primary decomposition is written in the user language.

Splitting field and Galois group computation \cite{ANY} are closely

related and are also important applications of polynomial

factorization.

\section{Techniques for efficient Groebner basis computation over {\bf Q}}

\label{gbtech}

In this section we review several practical techniques to improve

Groebner basis computation over {\bf Q}, which are easily

implemented but may not be well known.

Line 261 it practical. The following are major improvements:

Line 211 it practical. The following are major improvements:

\item Useless pair detection

We don't have to process all the pairs in $D$ and several useful

criteria for detecting useless pairs were proposed.

criteria for detecting useless pairs were proposed (cf. \cite{noro:BW}).

\item Selection strategy

The selection of $\{f,g\}$ greatly affects the subsequent computation.

The typical strategies are the normal startegy and the sugar strategy.

The typical strategies are the normal startegy \cite{noro:BUCH}

and the sugar strategy \cite{noro:SUGAR}.

The latter was proposed for efficient computation under a non

degree-compatible order.

Line 274 degree-compatible order.

Line 225 degree-compatible order.

Even if we apply several criteria, it is difficult to detect all pairs

whose S-polynomials are reduced to zero, and the cost to process them

often occupies a major part in the whole computation. The trace algorithms

often occupies a major part in the whole computation. The trace

were proposed to reduce such cost. This will be explained in more detail

algorithms \cite{noro:TRAV} were proposed to reduce such cost. This

in Section \ref{gbhomo}.

will be explained in more detail in Section \ref{sec:gbhomo}.

\item Change of ordering

For elimination, we need a Groebner basis with respect to a non

degree-compatible order, but it is often hard to compute it by

degree-compatible order, but it is often hard to compute it by the

the Buchberger algorithm. If the ideal is zero dimensional, we

Buchberger algorithm. If the ideal is zero dimensional, we can apply a

can apply a change of ordering algorithm for a Groebner basis

change of ordering algorithm \cite{noro:FGLM} for a Groebner basis

with respect to any order and we can obtain a Groebner basis

with respect to any order and we can obtain a Groebner basis with

with respect to a desired order.

respect to a desired order.

\end{itemize}

By implementing these techniques, one can obtain Groebner bases for

Line 294 problems with these classical tools. In the subsequent

Line 245 problems with these classical tools. In the subsequent

we show several methods for further improvements.

\subsection{Combination of homogenization and trace lifting}

\label{gbhomo}

\label{sec:gbhomo}

Traverso's trace lifting algorithm can be

The trace lifting algorithm can be

formulated in an abstract form as follows (c.f. \cite{FPARA}).

formulated in an abstract form as follows (c.f. \cite{noro:FPARA}).

\begin{tabbing}

Input : a finite subset $F \subset {\bf Z}[X]$\\

Output : a Groebner basis $G$ of $Id(F)$ with respect to a term order $<$\\

Line 336 lots of redundant elements can be removed.

Line 287 lots of redundant elements can be removed.

Let $I$ be a zero-dimensional ideal in $R={\bf Q}[x_1,\ldots,x_n]$.

Then the minimal polynomial $m(x_i)$ of a variable $x_i$ in $R/I$ can

be computed by a partial FGLM \cite{FGLM}, but it often takes long

be computed by a partial FGLM \cite{noro:FGLM}, but it often takes long

time if one searches $m(x_i)$ incrementally over {\bf Q}. In this

case we can apply a simple modular method to compute the minimal

polynomial.

Line 355 $GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once

Line 306 $GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once

candidate of $\deg(m(x_i))$, $m(x_i)$ can be determined by solving a

system of linear equations via the method of indeterminate

coefficient, and it can be solved efficiently by $p$-adic method.

Arguments on \cite{NOYO} ensures that $m(x_i)$ is what we want if it

Arguments on \cite{noro:NOYO} ensures that $m(x_i)$ is what we want if it

exists. Note that the full FGLM can also be computed by the same

method.

\subsection{Integer contents reduction}

\label{gbcont}

\label{sec:gbcont}

In some cases the cost to remove integer contents during normal form

computations is dominant. We can remove the content of an integral

polynomial $f$ efficiently by the following method \cite{REPL}.

polynomial $f$ efficiently by the following method \cite{noro:REPL}.

\begin{tabbing}

Input : an integral polynomial $f$\\

Output : a pair $(\cont(f),f/\cont(f))$\\

Line 386 $g_0$ with high accuracy. Then other components are ea

Line 337 $g_0$ with high accuracy. Then other components are ea

%cost for reading basis elements from disk is often negligible because

%of the cost for coefficient computations.

\section{Risa/Asir performance}

\subsection{Performances of Groebner basis computation}

We show timing data on Risa/Asir for Groebner basis computation

We show timing data on Risa/Asir for Groebner basis computation. The

and polynomial factorization. The measurements were made on

measurements were made on a PC with PentiumIII 1GHz and 1Gbyte of main

a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings

memory. Timings are given in seconds. In the tables `---' means it was

are given in seconds. In the tables `---' means it was not

not measured. $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura

measured.

$n$ system, both are famous bench mark problems \cite{noro:BENCH}. $McKay$

\cite{noro:REPL} is a system whose Groebner basis is hard to compute over

{\bf Q}. In Risa/Asir we have a test implemention of $F_4$ algorithm

\cite{noro:F4} and we also show its current performance. The term order is

graded reverse lexicographic order.

\subsection{Groebner basis computation}

Table \ref{tab:gbmod} shows timing data for Groebner basis computation

over $GF(32003)$. $F_4$ implementation in Risa/Asir outperforms

Buchberger algorithm implementation, but it is still several times

slower than $F_4$ implementation in FGb \cite{noro:FGB}.

Table \ref{gbmod} and Table \ref{gbq} show timing data for Groebner

Table \ref{tab:gbq} shows timing data for Groebner basis computation over

basis computation over $GF(32003)$ and over {\bf Q} respectively. In

$\Q$, where we compare the timing data under various configuration of

Table \ref{gbq} we compare the timing data under various configuration

algorithms. {\bf TR}, {\bf Homo}, {\bf Cont} means trace lifting,

of algorithms: with/without trace lifting, homogenization and contents

homogenization and contents reduction respectively.

reduction.

\ref{tab:gbq} also shows timings of minimal polynomial

$C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system,

computation for zero-dimensional ideals. Table \ref{tab:gbq} shows that

both are famous bench mark problems \cite{BENCH}. We also measured

it is difficult or practically impossible to compute Groebner bases of

the timing for $McKay$ system over {\bf Q} \cite{REPL}. the term

$C_7$, $C_8$ and $McKay$ without the methods described in Section

order is graded reverse lexicographic order.

\ref{sec:gbhomo} and \ref{sec:gbcont}.

As to the Buchberger algorithm over $GF(32003)$,

Singular\cite{SINGULAR}'s Buchberger algorithm implementation shows

better performance than Risa/Asir. $F_4$ implementation in Risa/Asir

is outperforms both of them, but it is still several times slower than

$F_4$ implementation in FGb \cite{FGB}.

Table \ref{gbq} shows that it is difficult or practically impossible

to compute Groebner bases of $C_7$, $C_8$ and $McKay$ without

the methods described in Section \ref{gbhomo} and \ref{gbcont}.

Though $F_4$ implementation in Risa/Asir over {\bf Q} is still

experimental, the timing of $McKay$ is greatly reduced.

Fig. \ref{f4vsbuch} explains why $F_4$ is efficient in this case. The

Fig. \ref{tab:f4vsbuch} explains why $F_4$ is efficient in this case. The

figure shows that the Buchberger algorithm produces normal forms with

huge coefficients for S-polynomials after the 250-th one, which are

the computations in degree 16. However, we know that the reduced

Line 437 FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \

Line 385 FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \

\end{tabular}

\end{center}

\caption{Groebner basis computation over $GF(32003)$}

\label{gbmod}

\label{tab:gbmod}

\end{table}

\begin{table}[hbtp]

\begin{center}

\begin{tabular}{|c||c|c|c|c|c|} \hline

& $C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline

TR+Homo,Cont & 389 & 54000 & 29 & 299 & 34950 \\ \hline

TR+Homo+Cont & 389 & 54000 & 29 & 299 & 34950 \\ \hline

TR+Homo & --- & --- & --- & --- & --- \\ \hline

TR & --- & --- & --- & --- & --- \\ \hline

TR & --- & --- & --- & --- & --- \\ \hline \hline

Minipoly & --- & --- & --- & --- & N/A \\ \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 & 288 & 0.6 & 5 & 10 \\ \hline

\end{tabular}

(TR : trace lifting; Homo : homogenization; Cont : contents reduction)

\end{center}

\caption{Groebner basis computation over {\bf Q}}

\caption{Groebner basis and minimal polynomial computation over {\bf Q}}

\label{gbq}

\label{tab:gbq}

\end{table}

\begin{figure}[hbtp]

Line 466 TR & --- & --- & --- & --- & --- \\ \hline

Line 412 TR & --- & --- & --- & --- & --- \\ \hline

\epsffile{blen.ps}

\end{center}

\caption{Maximal coefficient bit length of intermediate bases}

\label{f4vsbuch}

\label{tab:f4vsbuch}

\end{figure}

Table \ref{minipoly} shows timing data for the minimal polynomial

%Table \ref{minipoly} shows timing data for the minimal polynomial

computations of all variables over {\bf Q} by the modular method.

%computations of all variables over {\bf Q} by the modular method.

\begin{table}[hbtp]

%\begin{table}[hbtp]

\begin{center}

%\begin{center}

\begin{tabular}{|c||c|c|c|c|c|} \hline

%\begin{tabular}{|c||c|c|c|c|c|} \hline

& $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline

% & $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline

%Singular & 0.9 & 846 & 307 & 60880 & --- \\ \hline

Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

%Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

\end{tabular}

%\end{tabular}

\end{center}

%\end{center}

\caption{Minimal polynomial computation}

%\caption{Minimal polynomial computation}

\label{minipoly}

%\label{minipoly}

\end{table}

%\end{table}

\subsection{Polynomial factorization}

%\subsection{Polynomial factorization}

%Table \ref{unifac} shows timing data for univariate factorization over

%{\bf Q}. $N_{i,j}$ is an irreducible polynomial which are hard to

%factor by the classical algorithm. $N_{i,j}$ is a norm of a polynomial

%and $\deg(N_i) = i$ with $j$ modular factors. Risa/Asir is

%disadvantageous in factoring polynomials of this type because the

%algorithm used in Risa/Asir has exponential complexity. In contrast,

%CoCoA 4\cite{COCOA} and NTL-5.2\cite{NTL} show nice performances

%CoCoA 4\cite{noro:COCOA} and NTL-5.2\cite{noro:NTL} show nice performances

%because they implement recently developed algorithms.

%\begin{table}[hbtp]

Line 508 Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

Line 454 Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline

%\caption{Univariate factorization over {\bf Q}}

%\label{unifac}

%\end{table}

Table \ref{multifac} shows timing data for multivariate factorization

%Table \ref{multifac} shows timing data for multivariate factorization

over {\bf Q}. $W_{i,j,k}$ is a product of three multivariate

%over {\bf Q}. $W_{i,j,k}$ is a product of three multivariate

polynomials $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file {\tt

%polynomials $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file {\tt

fctrdata} in Asir library directory. It is also included in Risa/Asir

%fctrdata} in Asir library directory. It is also included in Risa/Asir

source tree and located in {\tt asir2000/lib}. These examples have

%source tree and located in {\tt asir2000/lib}. These examples have

leading coefficients of large degree which vanish at 0 which tend to

%leading coefficients of large degree which vanish at 0 which tend to

cause so-called the leading coefficient problem the bad zero

%cause so-called the leading coefficient problem the bad zero

problem. Risa/Asir's implementation carefully treats such cases and it

%problem. Risa/Asir's implementation carefully treats such cases and it

shows reasonable performance compared with other famous systems.

%shows reasonable performance compared with other famous systems.

\begin{table}[hbtp]

%\begin{table}[hbtp]

\begin{center}

%\begin{center}

\begin{tabular}{|c||c|c|c|c|c|} \hline

%\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

% & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline

variables & 3 & 5 & 5 & 5 & 4 \\ \hline

%variables & 3 & 5 & 5 & 5 & 4 \\ \hline

monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline

%monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline

Asir & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \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

%Mathematica 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline

Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline

%Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline

\end{tabular}

%\end{tabular}

\end{center}

%\end{center}

\caption{Multivariate factorization over {\bf Q}}

%\caption{Multivariate factorization over {\bf Q}}

\label{multifac}

%\label{multifac}

\end{table}

%\end{table}

As to univariate factorization over {\bf Q}, the univariate factorizer

%As to univariate factorization over {\bf Q}, the univariate factorizer

implements old algorithms and its behavior is what one expects,

%implements old algorithms and its behavior is what one expects,

that is, it shows average performance in cases where there are little

%that is, it shows average performance in cases where there are little

extraneous factors, but shows poor performance for hard to factor

%extraneous factors, but shows poor performance for hard to factor

polynomials with many extraneous factors.

%polynomials with many extraneous factors.

\section{OpenXM and Risa/Asir OpenXM interfaces}

Line 546 polynomials with many extraneous factors.

Line 492 polynomials with many extraneous factors.

OpenXM stands for Open message eXchange protocol for Mathematics.

From the viewpoint of protocol design, it can be regarded as a child

of OpenMath \cite{OPENMATH}. However our approach is somewhat

of OpenMath \cite{noro:OPENMATH}. However our approach is somewhat

different. Our main purpose is to provide an environment for

integrating {\it existing} mathematical software systems. OpenXM

RFC-100 \cite{RFC100} defines a client-server architecture. Under

RFC-100 \cite{noro:RFC100} defines a client-server architecture. Under

this specification, a client invokes an OpenXM ({\it OX}) server. The

client can send OpenXM ({\it OX}) messages to the server. OX messages

consist of {\it data} and {\it command}. Data is encoded according to

Line 566 hybrid server.

Line 512 hybrid server.

OpenXM RFC-100 also defines methods for session management. In particular

the method to reset a server is carefully designed and it provides

a robust way of using servers both for interactive and non-interactive

purposes.

\subsection{OpenXM API in Risa/Asir user language}

Line 662 readable form, which may be sufficient for a simple us

Line 608 readable form, which may be sufficient for a simple us

interface.

\section{Concluding remarks}

We have shown the current status of Risa/Asir and its OpenXM

%We have shown the current status of Risa/Asir and its OpenXM

interfaces. As a result of our policy of development, it is true that

%interfaces. As a result of our policy of development, it is true that

Risa/Asir does not have abundant functions. However it is a completely

%Risa/Asir does not have abundant functions. However it is a completely

open system and its total performance is not bad. Especially on

%open system and its total performance is not bad. Especially on

Groebner basis computation over {\bf Q}, many techniques for improving

%Groebner basis computation over {\bf Q}, many techniques for improving

practical performances have been implemented. As the OpenXM interface

%practical performances have been implemented. As the OpenXM interface

specification is completely documented, we can easily add another

%specification is completely documented, we can easily add another

function to Risa/Asir by wrapping an existing software system as an OX

%function to Risa/Asir by wrapping an existing software system as an OX

server, and other clients can call functions in Risa/Asir by

%server, and other clients can call functions in Risa/Asir by

implementing the OpenXM client interface. With the remote debugging

%implementing the OpenXM client interface. With the remote debugging

and the function to reset servers, one will be able to enjoy parallel

%and the function to reset servers, one will be able to enjoy parallel

and distributed computation with OpenXM facilities.

%and distributed computation with OpenXM facilities.

We have shown that many techniques for

improving practical performances are implemented in Risa/Asir's

Groebner basis engine. Though another important function, the

polynomial factorizer only implements classical algorithms, its

performance is comparable with or superior to that of Maple or

Mathematica and is still practically useful. By preparing OpenXM

interface or simply linking the Asir OpenXM library, one can call

these efficient functions from any application. Risa/Asir is a

completely open system. It is open source software

and the OpenXM interface specification is completely documented, one

can easily write interfaces to call functions in Risa/Asir and one

will be able to enjoy parallel and distributed computation.

\begin{thebibliography}{7}

\addcontentsline{toc}{section}{References}

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Anay, H., Noro, M., Yokoyama, K. (1996)

Computation of the Splitting fields and the Galois Groups of Polynomials.

Algorithms in Algebraic geometry and Applications,

Birkh\"auser (Proceedings of MEGA'94), 29--50.

\bibitem{FPARA}

\bibitem{noro:FPARA}

Jean-Charles Faug\`ere (1994)

Parallelization of Groebner basis.

Proceedings of PASCO'94, 124--132.

\bibitem{F4}

\bibitem{noro:F4}

Jean-Charles Faug\`ere (1999)

A new efficient algorithm for computing Groebner bases ($F_4$).

Journal of Pure and Applied Algebra (139) 1-3 , 61--88.

\bibitem{FGLM}

\bibitem{noro:FGLM}

Faug\`ere, J.-C. et al. (1993)

Efficient computation of zero-dimensional Groebner bases by change of ordering.

Journal of Symbolic Computation 16, 329--344.

\bibitem{RFC100}

\bibitem{noro:RFC100}

M. Maekawa, et al. (2001)

The Design and Implementation of OpenXM-RFC 100 and 101.

Proceedings of ASCM2001, World Scientific, 102--111.

\bibitem{RISA}

\bibitem{noro:RISA}

Noro, M. et al. (1994-2001)

A computer algebra system Risa/Asir.

{\tt http://www.openxm.org}, {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}.

\bibitem{REPL}

\bibitem{noro:REPL}

Noro, M., McKay, J. (1997)

Computation of replicable functions on Risa/Asir.

Proceedings of PASCO'97, ACM Press, 130--138.

\bibitem{NOYO}

\bibitem{noro:NOYO}

Noro, M., Yokoyama, K. (1999)

A Modular Method to Compute the Rational Univariate

Representation of Zero-Dimensional Ideals.

Journal of Symbolic Computation, 28, 1, 243--263.

\bibitem{OPENXM}

\bibitem{noro:OPENXM}

OpenXM committers (2000-2001)

OpenXM package.

{\tt http://www.openxm.org}.

\bibitem{RUR}

\bibitem{noro:RUR}

Rouillier, R. (1996)

R\'esolution des syst\`emes z\'ero-dimensionnels.

Doctoral Thesis(1996), University of Rennes I, France.

\bibitem{SY}

\bibitem{noro:SY}

Shimoyama, T., Yokoyama, K. (1996)

Localization and Primary Decomposition of Polynomial Ideals.

Journal of Symbolic Computation, 22, 3, 247--277.

\bibitem{TRAGER}

\bibitem{noro:TRAGER}

Trager, B.M. (1976)

Algebraic Factoring and Rational Function Integration.

Proceedings of SYMSAC 76, 219--226.

\bibitem{TRAV}

\bibitem{noro:TRAV}

Traverso, C. (1988)

Groebner trace algorithms.

LNCS {\bf 358} (Proceedings of ISSAC'88), Springer-Verlag, 125--138.

\bibitem{BENCH}

\bibitem{noro:BENCH}

{\tt http://www.math.uic.edu/\~\,jan/demo.html}.

\bibitem{COCOA}

\bibitem{noro:COCOA}

{\tt http://cocoa.dima.unige.it/}.

\bibitem{FGB}

\bibitem{noro:FGB}

{\tt http://www-calfor.lip6.fr/\~\,jcf/}.

%\bibitem{NTL}

%\bibitem{noro:NTL}

%{\tt http://www.shoup.net/}.

\bibitem{OPENMATH}

\bibitem{noro:OPENMATH}

{\tt http://www.openmath.org/}.

\bibitem{SINGULAR}

\bibitem{noro:SINGULAR}

{\tt http://www.singular.uni-kl.de/}.

\end{thebibliography}

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