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Line 282 such that $HT(h)|HT(g)$ \}

The input is homogenized to suppress intermediate coefficient swells

of intermediate basis elements. The homogenization may increase the

number of normal forms reduced to zero, but they can be

detected over by the computations over $GF(p)$. Finally, by

detected by the computations over $GF(p)$. Finally, by

dehomogenizing the candidate we can expect that lots of redundant

elements are removed and the subsequent check are made easy.

Line 342 $g_0$ with high accuracy. Then other components are ea

\subsection{Performances of Groebner basis computation}

All the improvements in this sections have been implemented in

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

All the improvements in this section have been implemented in

Risa/Asir. Besides we have a test implemention of $F_4$ algorithm

\cite{noro:F4}, which is a new algorithm for computing Groebner basis

\cite{noro:F4}, which is a new algorithm for computing Groebner basis.

by various methods. We show timing data on Risa/Asir for Groebner

The measurements were made on a PC with PentiumIII

basis computation. The measurements were made on a PC with PentiumIII

1GHz and 1Gbyte of main memory. Timings are given in seconds. In the

tables `exhasut' means memory exhastion. $C_n$ is the cyclic $n$

tables `exhaust' means memory exhastion. $C_n$ is the cyclic $n$

system and $K_n$ is the Katsura $n$ 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}. The term order