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% $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.5 2001/11/28 08:46:54 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.

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\usepackage{epsfig}

\def\cont{{\rm cont}}

\def\GCD{{\rm GCD}}

\def\Q{{\bf Q}}

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

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% allows abbreviation of title, if the full title is too long

% to fit in the running head

\author{Masayuki Noro\inst{1}}

\author{Masayuki Noro}

%\authorrunning{Masayuki Noro}

% if there are more than two authors,

Line 87

Line 88

\maketitle % typesets the title of the contribution

\begin{abstract}

%\begin{abstract}

OpenXM \cite{OPENXM} is an infrastructure for exchanging mathematical

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

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

%developed for testing experimental polynomial algorithms, and now it

distributed computation. Risa/Asir is software for polynomial

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

computation. It has been developed for testing new algorithms, and now

%OpenXM is an infrastructure for exchanging mathematical

it acts as both a client and a server in the OpenXM package. In this

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

article we present an overview of Risa/Asir and review several

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

techniques for improving performances of Groebner basis computation.

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

We also show Risa/Asir's OpenXM interfaces and their usages by

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

examples.

%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. It is open source and the source code

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

and binaries are available via {\tt ftp} or {\tt CVS}. Risa/Asir is

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

not only a standalone computer algebra system but also a main

Risa/Asir has been used for implementing and testing experimental

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

algorithms such as polynomial factorizations, splitting field and

various software compliant to OpenXM protocol specification. OpenXM

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

is an infrastructure for exchanging mathematical data and Risa/Asir

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

has three kind of OpenXM interfaces : client interfaces, an OpenXM

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

server, and a subroutine library. Our goals of developing Risa/Asir

funtions are polynomial factorization and Groebner basis

are as follows:

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

fundamentals and vaious efforts for improving efficiency especially

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

In this section we review several practical techniques to improve

polynomial factorization, computation related to Groebner basis,

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

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

implemented but may not be well known.

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

We use the following notations.

various modular techniques to overcome difficulties caused by huge

\begin{description}

intermediate coefficients. We have had several results and they have

\item $<$ : a term order in the set of monomials. It is a total order such that

been implemented in Risa/Asir.

\item General purpose open system

$\forall t, 1 \le t$ and $\forall s, t, u, s<t \Rightarrow us<ut$.

\item $Id(F)$ : a polynomial ideal generated by a polynomial set $F$.

\item $HT(f)$ : the head term of a polynomial with respect to a term order.

\item $HC(f)$ : the head coefficient of a polynomial with respect to a term order.

\item $T(f)$ : terms with non zero coefficients in $f$.

\item $Spoly(f,g)$ : the S-polynomial of $\{f,g\}$

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

$Spoly(f,g) = T_{f,g}/HT(f)\cdot f/HC(f) -T_{f,g}/HT(g)\cdot g/HC(g)$, where

computer algebra system. In recent years we can obtain various high

$T_{f,g} = LCM(HT(f),HT(g))$.

performance applications or libraries as free software. We wrapped

\item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$.

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

\end{description}

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

\subsection{Groebner basis computation and its improvements}

The origin of OpenXM is a protocol for doing parallel distributed

A Groebner basis of an ideal $Id(F)$ can be computed by the Buchberger

computations by connecting multiple Risa/Asir's over TCP/IP. OpenXM is

algorithm. The key oeration in the algorithm is the following

also designed to provide an environment efficient parallel distributed

division by a polynomial set.

computation. Currently only client-server communication is available,

\begin{tabbing}

but we are preparing a specification OpenXM-RFC 102 allowing

while \= $\exists g \in G$, $\exists t \in T(f)$ such that $HT(g)|t$ do\\

client-client communication, which will enable us to execute wider

\> $f \leftarrow f - t/HT(g) \cdot c/HC(g) \cdot g$, \quad

range of parallel algorithms efficiently.

where $c$ is the coeffcient of $t$ in $f$

\end{enumerate}

\end{tabbing}

This division terminates for any term order.

With this division, we can show the most primitive version of the

Buchberger algorithm.

\begin{tabbing}

Input : a finite polynomial set $F$\\

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

$G \leftarrow F$; \quad $D \leftarrow \{\{f,g\}| f, g \in G, f \neq g \}$\\

while \= $D \neq \emptyset$ do \\

\> $\{f,g\} \leftarrow$ an element of $D$; \quad

$D \leftarrow D \setminus \{P\}$\\

\> $R \leftarrow$ a remainder of $Spoly(f,g)$ on division by $G$\\

\> if $R \neq 0$ then $D \leftarrow D \cup \{\{f,R\}| f \in G\}$; \quad

$G \leftarrow G \cup \{R\}$\\

end do\\

return G

\end{tabbing}

Though this algorithm gives a Groebner basis of $Id(F)$,

it is not practical at all. We need lots of techniques to make

it practical. The following are major improvements:

\begin{itemize}

\item Useless pair detection

\subsection{Groebner basis and the related computation}

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

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

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

\item Selection strategy

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 the rationals, 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 the rationals. 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 is hard to compute without

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

of ordering algorithm and rational univariate representation. We also

made a test implementation of $F_4$ algorithm \cite{F4}. Later we will

show timing data on Groebner basis computation.

\subsection{Polynomial factorization}

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

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.

Here we briefly review functions on polynomial factorization. For

\item Modular methods

univariate factorization over {\bf Q}, the classical

Berlekamp-Zassenhaus algorithm is implemented. Efficient algorithms

recently proposed have not yet implemented. For Univariate factorizer

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 the

rationals \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 the rationals,

the classical EZ(Extended Zassenhaus) type algorithm is implemented.

\subsection{Other functions}

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

By applying Groebner basis computation and polynomial factorization,

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

we have implemented several higher level algorithms. A typical

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

application is primary ideal decomposition of polynomial ideals over

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

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

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

\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}}

\item Change of ordering

\label{gbtech}

In this section we review several practical techniques to improve

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

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

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

implemented but may not be well known.

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

We use the following notations.

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

\begin{description}

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

\item $Id(F)$ : a polynomial ideal generated by $F$

respect to a desired order.

\item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$

\item $HT(f)$ : the head term of a polynomial with respect to a term order

\item $HC(f)$ : the head coefficient of a polynomial with respect to a term order

\end{description}

\end{itemize}

By implementing these techniques, one can obtain Groebner bases for

wider range of inputs. Nevertheless there are still intractable

problems with these classical tools. In the subsequent sections

we show several methods for further improvements.

\subsection{Combination of homogenization and trace lifting}

\label{sec:gbhomo}

Traverso's trace lifting algorithm can be

The trace lifting algorithm can be

formulated in an abstract form as follows \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 252 The input is homogenized to suppress intermediate coef

Line 281 The input is homogenized to suppress intermediate coef

of intermediate basis elements. The number of zero normal forms may

increase by the homogenization, but they are detected over

$GF(p)$. Finally, by dehomogenizing the candidate we can expect that

lots of redundant elements can be removed. We will show later that this is

lots of redundant elements can be removed.

surely efficient for some input polynomial sets.

\subsection{Minimal polynomial computation by modular method}

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.

\begin{tabbing}

Input : a Groebner basis $G$ of $I$, a variable $x_i$\\

Output : the minimal polynomial of $x$ in $R/I$\\

Output : the minimal polynomial of $x_i$ in $R/I$\\

do \= \\

\> $p \leftarrow$ a new prime such that $p \not{|} HC(g)$ for all $g \in G$\\

\> $m_p \leftarrow$ the minimal polynomial of $x_i$ in $GF(p)[x_1,\ldots,x_n]/Id(\phi_p(G))$\\

Line 276 In this algorithm, $m_p$ can be obtained by a partial

Line 305 In this algorithm, $m_p$ can be obtained by a partial

$GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once we know the

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

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

want if it exists. Note that the full FGLM can also be computed by the

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

same method.

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

method.

\subsection{Integer contents reduction}

\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 $(\cont(f),f/\cont(f))$\\

$g_0 \leftarrow$ an estimate of $\cont(f)$ such that $\cont(f)|g_0$\\

Write $f$ as $f = g_0q+r$ by division with remainder for each coefficient\\

Write $f$ as $f = g_0q+r$ by division with remainder by $g_0$ for each coefficient\\

If $r = 0$ then return $(g_0,q)$\\

else return $(g,g_0/g \cdot q + r/g)$, where $g = \GCD(g_0,\cont(r))$

\end{tabbing}

By separating the set of coefficients of $f$ into two subsets and by

computing GCD of sums in the elements in the subsets we can estimate

computing GCD of sums of the elements in each subset we can estimate

$g_0$ with high accuracy. Then other components are easily computed.

%\subsection{Demand loading of reducers}

Line 306 $g_0$ 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.

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

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

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

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

homogenization and contents reduction respectively.

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

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

order is graded reverse lexicographic order. In the both tables, the

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

first three rows are timings for the Buchberger algorithm, and the

it is difficult or practically impossible to compute Groebner bases of

last two rows are timings for $F_4$ algorithm. As to the Buchberger

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

algorithm over $GF(32003)$, Singular\cite{SINGULAR} shows the best

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

performance among the three systems. $F_4$ implementation in Risa/Asir

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

is faster than the Buchberger algorithm implementation in Singular,

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

but it is still several times slower than $F_4$ implementation in FGb

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

\cite{FGB}. In Table \ref{gbq}, $C_7$ and $McKay$ can be computed by

figure shows that the Buchberger algorithm produces normal forms with

the Buchberger algorithm with the methods described in Section

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

\ref{gbtech}. It is obvious that $F_4$ implementation in Risa/Asir

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

over {\bf Q} is too immature. Nevertheless the timing of $McKay$ is

basis elements have much smaller coefficients after removing contents.

greatly reduced. Fig. \ref{f4vsbuch} explains why $F_4$ is efficient

As $F_4$ algorithm automatically produces the reduced ones, the degree

in this case. The figure shows that the Buchberger algorithm produces

16 computation is quite easy in $F_4$.

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 basis elements have much smaller coefficients after

removing contents. As $F_4$ algorithm automatically produces the

reduced ones, the degree 16 computation is quite easy in $F_4$.

\begin{table}[hbtp]

\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

%Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline

CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\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}

\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|c|} \hline

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

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

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

Asir $Buchberger$ & 389 & 594 & 54000 & 29 & 299 & 34950 \\ \hline

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

Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline

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

CoCoA 4 & --- & 13227 & --- & 57 & 709 & --- \\ \hline\hline

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

Asir $F_4$ & 989 & 456 & --- & 90 & 991 & 4939 \\ \hline

Minipoly & --- & --- & --- & --- & N/A \\ \hline

FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \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}

\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 377 FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \hline

Line 412 FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \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

computation over {\bf Q}. Singular provides a function {\tt finduni}

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

for computing the minimal polynomial in each variable in ${\bf

%\begin{table}[hbtp]

Q}[x_1,\ldots,x_n]/I$ for zero dimensional ideal $I$. The modular

%\begin{center}

method used in Asir is efficient when the resulting minimal

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

polynomials have large coefficients and we can verify the fact from Table

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

\ref{minipoly}.

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

\begin{table}[hbtp]

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

\begin{center}

%\end{tabular}

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

%\end{center}

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

%\caption{Minimal polynomial computation}

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

%\label{minipoly}

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

%\end{table}

\end{tabular}

\end{center}

\caption{Minimal polynomial computation}

\label{minipoly}

\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 424 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

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

factorization over {\bf Q}.

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

$W_{i,j,k}$ is a product of three multivariate polynomials

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

$Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file

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

{\tt fctrdata} in Asir library directory. It is also included

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

in Risa/Asir source tree and located in {\tt asir2000/lib}.

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

For these examples Risa/Asir shows reasonable performance

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

compared with other famous systems.

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

\begin{table}[hbtp]

%shows reasonable performance compared with other famous systems.

\begin{center}

%\begin{table}[hbtp]

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

%\begin{center}

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

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

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

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

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

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

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

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

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

%Singular& $>$15min & --- & ---& ---& ---\\ \hline

CoCoA 4 & 5.2 & $>$15min & $>$15min & $>$15min & 117 \\ \hline\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},

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

the univariate factorizer implements only classical

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

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

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

where there are little extraneous factors, but

%polynomials with many extraneous factors.

shows poor performance for hard to factor polynomials.

\section{OpenXM and Risa/Asir OpenXM interfaces}

Line 462 shows poor performance for hard to factor polynomials.

Line 492 shows poor performance for hard to factor polynomials.

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 482 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 client interface of {\tt asir}}

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

Risa/Asir is a main client in OpenXM package. The application {\tt

asir} can access to OpenXM servers via several built-in interface

Line 504 We show a typical OpenXM session.

Line 534 We show a typical OpenXM session.

[[1,1],[x^4+y*x^3+y^2*x^2+y^3*x+y^4,1],

[x^4-y*x^3+y^2*x^2-y^3*x+y^4,1],[x-y,1],[x+y,1]]

[5] ox_cmo_rpc(P,"fctr,",x^10000-2^10000*y^10000);

/* call factorizer; an utility function */

/* call factorizer; a utility function */

[6] ox_reset(P); /* reset the computation in the server */

Line 516 We show a typical OpenXM session.

Line 546 We show a typical OpenXM session.

An application {\tt ox\_asir} is a wrapper of {\tt asir} and provides

all the functions of {\tt asir} to OpenXM clients. It completely

implements the OpenXM reset protocol and also provides remote

implements the OpenXM reset protocol and also allows remote

debugging of user programs running on the server. As an example we

show a program for checking whether a polynomial set is a Groebner

basis or not. A client executes {\tt gbcheck()} and servers execute

{\tt sp\_nf\_for\_gbcheck()} which is a simple normal form computation

of a S-polynomial. First of all the client collects all critical pairs

of an S-polynomial. First of all the client collects all critical pairs

necessary for the check. Then the client requests normal form

computations to idling servers. If there are no idling servers the

clients waits for some servers to return results by {\tt

Line 563 def gbcheck(B,V,O,Procs) {

Line 593 def gbcheck(B,V,O,Procs) {

}

\end{verbatim}

\subsection{Asir OpenXM library {\tt libasir.a}}

\subsection{OpenXM C language API in {\tt libasir.a}}

Asir OpenXM library {\tt libasir.a} includes functions simulating the

Risa/Asir subroutine library {\tt libasir.a} contains functions

stack machine commands supported in {\tt ox\_asir}. By linking {\tt

simulating the stack machine commands supported in {\tt ox\_asir}. By

libasir.a} an application can use the same functions as in {\tt

linking {\tt libasir.a} an application can use the same functions as

ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is

in {\tt ox\_asir} without accessing to {\tt ox\_asir} via

also a stack, which can be manipulated by library functions. In

TCP/IP. There is also a stack, which can be manipulated by the library

order to make full use of this interface, one has to prepare

functions. In order to make full use of this interface, one has to

conversion functions between CMO and the data structures proper to the

prepare conversion functions between CMO and the data structures

application. A function {\tt asir\_ox\_pop\_string()} is provided to

proper to the application itself. A function {\tt

convert CMO to a human readable form, which may be sufficient for a

asir\_ox\_pop\_string()} is provided to convert CMO to a human

simple use of this interface.

readable form, which may be sufficient for a simple use of this

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 vice versa. User program debugger can be used both for

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

local and remote debugging. By combining the debugger and the function

%implementing the OpenXM client interface. With the remote debugging

to reset servers, one will be able to enjoy parallel and distributed

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

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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Jean-Charles Faug\`ere (1994)

Parallelization of Groebner basis.

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\bibitem{F4}

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Jean-Charles Faug\`ere (1999)

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

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%\bibitem{NTL}

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

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