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version 1.8, 2000/01/15 06:26:06 version 1.9, 2000/01/16 06:39:39
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 % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.7 2000/01/15 06:11:17 takayama Exp $  % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.8 2000/01/15 06:26:06 takayama Exp $
 \section{Applications}  \section{Applications}
   
 \subsection{Heterogeneous Servers}  \subsection{Heterogeneous Servers}
Line 36  His algorithm reduces the problem to computations of G
Line 36  His algorithm reduces the problem to computations of G
 in $D$ and to find the maximal integral root of a polynomial.  in $D$ and to find the maximal integral root of a polynomial.
 This algorithm (the function {\tt annfs}) is implemented by  This algorithm (the function {\tt annfs}) is implemented by
 kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and  kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
 ox\_asir, to factorize polynomials to find the integral  {\tt ox\_asir}, to factorize polynomials to find the integral
 roots.  roots.
 These two OpenXM complient systems are integrated by  These two OpenXM complient systems are integrated by
 OpenXM protocol.  OpenXM protocol.
Line 63  an algorithm to stratify singularity 
Line 63  an algorithm to stratify singularity 
 \cite{oaku-advance}  \cite{oaku-advance}
 is implemented by  is implemented by
 kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and  kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
 ox\_asir, for primary ideal decompositions.  {\tt ox\_asir}, for primary ideal decompositions.
   
 \subsubsection{A Course on Solving Algebraic Equations}  \subsubsection{A Course on Solving Algebraic Equations}
   
 Risa/asir \cite{asir} is a general computer algebra system  Risa/Asir \cite{asir} is a general computer algebra system
 which is good at Gr\"obner basis computations for zero dimensional ideal  which can be used for Gr\"obner basis computations for zero dimensional ideal
 with ${\bf Q}$ coefficients.  with ${\bf Q}$ coefficients.
 However, it is not good at graphical presentations and  However, it is not good at graphical presentations and
 numerical methods.  numerical methods.
 We integrated Risa/asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}  We integrated Risa/Asir, ox\_phc (based on PHC pack by Verschelde \cite{phc}
 for the polyhedral homotopy method) and  for the polyhedral homotopy method) and
 ox\_gnuplot (GNUPLOT) servers  ox\_gnuplot (GNUPLOT) servers
 to teach a course on solving algebraic equations.  to teach a course on solving algebraic equations.
 This course was presented with the text book \cite{CLO} which discusses  This course was presented with the text book \cite{CLO},
   which discusses
 on the Gr\"obner basis method and the polyhedral homotopy method  on the Gr\"obner basis method and the polyhedral homotopy method
 to solve systems of algebraic equations.  to solve systems of algebraic equations.
 We could teach a course  We could teach a course
 with a unified environment  with a unified environment
 controlled by asir user language, which is similar to C.  controlled by Asir user language, which is similar to C.
 The following is an asir session to solve algebraic equations by calling  The following is an Asir session to solve algebraic equations by calling
 the PHC pack (see Figure \ref{katsura} too):  the PHC pack (Figure \ref{katsura} is the output of {\tt [292]}):
 \begin{verbatim}  \begin{verbatim}
 [287] phc(katsura(7));  [287] phc(katsura(7));
 The detailed output is in the file tmp.output.*  The detailed output is in the file tmp.output.*
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