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version 1.7, 2000/01/15 06:11:17 version 1.12, 2000/01/17 08:06:15
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 % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.6 2000/01/15 03:47:59 takayama Exp $  % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.11 2000/01/17 07:06:53 noro Exp $
 \section{Applications}  \section{Applications}
   
 \subsection{Heterogeneous Servers}  \subsection{Heterogeneous Servers}
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 By using OpenXM, we can treat OpenXM servers essentially  By using OpenXM, we can treat OpenXM servers essentially
 like a subroutine.  like a subroutine.
 Since OpenXM provides a universal stackmachine which does not  Since OpenXM provides a universal stack machine which does not
 depend each servers,  depend each servers,
 it is relatively easy to install new servers.  it is relatively easy to install new servers.
 We can build a new computer math system by assembling  We can build a new computer math system by assembling
 different OpenXM servers.  different OpenXM servers.
 It is similar to building a toy house by LEGO blocks.  It is similar to building a toy house by LEGO blocks.
   
 We will see two examples of custom made systems  We will see two examples of custom-made systems
 built by OpenXM servers.  built by OpenXM servers.
   
 \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}  \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
Line 33  equations for the function $f^{-1}$.
Line 33  equations for the function $f^{-1}$.
 An algorithm to determine generators of the annihilating ideal  An algorithm to determine generators of the annihilating ideal
 was given by Oaku (see, e.g., \cite[5.3]{sst-book}).  was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
 His algorithm reduces the problem to computations of Gr\"obner bases  His algorithm reduces the problem to computations of Gr\"obner bases
 in $D$ and to find the maximal integral root of a polynomial.  in $D$ and to find the minimal 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 compliant systems are integrated by
 OpenXM protocol.  the OpenXM protocol.
   
 For example, the following is a sm1 session to find the annihilating  For example, the following is a sm1 session to find the annihilating
 ideal for $f = x^3 - y^2 z^2$.  ideal for $f = x^3 - y^2 z^2$.
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 taught the 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.  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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