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version 1.3, 2000/01/03 04:27:52 version 1.4, 2000/01/15 01:33:32
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 % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.2 2000/01/02 07:32:11 takayama Exp $  % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.3 2000/01/03 04:27:52 takayama Exp $
   
 \subsection{Heterotic Network}   (Takayama)  \subsection{Heterogeneous Servers}
   
 \def\pd#1{ \partial_{#1} }  \def\pd#1{ \partial_{#1} }
 \subsubsection{Annihilating ideal}  
   
   By using OpenXM, we can treat OpenXM servers essentially
   like a subroutine.
   Since OpenXM provides a universal stackmachine which does not
   depend each servers,
   it is relatively easy to install new servers.
   We can build a new computer math system by assembling
   different OpenXM servers.
   It is similar to building a toy house by LEGO blocks.
   
   We will see two examples of special purpose systems
   built by OpenXM servers.
   
   \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
   
 Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$  Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
 be the ring of differential operators.  be the ring of differential operators.
 For a given polynomial  For a given polynomial
Line 17  Here, $\bullet$ denotes the action of $D$ to functions
Line 30  Here, $\bullet$ denotes the action of $D$ to functions
 The annihilating ideal can be regarded as the maximal differential  The annihilating ideal can be regarded as the maximal differential
 equations for the function $f^{-1}$.  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 in 1995 (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 maximal integral root of a polynomial.
 An implementation of this algorithm (the function {\tt annfs})  This algorithm (the function {\tt annfs}) is implemented by
 on kan/sm1 \cite{kan}  kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
 calls ox\_asir to factorize polynomials to find the integral  ox\_asir, to factorize polynomials to find the integral
 roots.  roots.
   These two OpenXM complient systems are integrated by
   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 $f = x^3 - y^2 z^2$.
 \begin{verbatim}  \begin{verbatim}
Line 41  $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
Line 57  $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
 and the minimal integral root is $-1$  and the minimal integral root is $-1$
 as shown in the output.  as shown in the output.
   
 \subsubsection{Primary ideal decomposition and  Similarly,
 a stratification of singularity}  an algorithm to stratify singularity
 Let ${\cal D} = {\cal O}\langle \pd{1}, \ldots, \pd{n} \rangle$  \cite{oaku-advance}
 be the sheaf of algebraic differential operators on ${\bf C}^n$.  is implemented by
 For a given polynomial  kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
 $f \in {\bf Q}[x_1, \ldots, x_n]$,  ox\_asir, for primary ideal decompositions.
 minimal degree non-zero polynomial $b_a(s)$ such that  
 $ b_a(s) f^s \in {\cal D}_{a} \bullet f^{s+1} $  
 is called the local $b$-function of $f^s$ at  
 $x=a$.  
 Here, ${\cal D}_a$ denotes the germ of the sheaf ${\cal D}$ at  
 $x=a$.  
 The polynomial $b_a(s)$ depends on $a$ and ${\bf C}^n$ is  
 stratified into a finite algebraic sets such that  
 $b_a(s)$ is a constant with respect to $a$ on each  
 algebraic set.  
 An algorithm to determine the stratification  
 was given by Oaku in 1997 \cite{oaku-advance}.  
 His algorithm reduces the problem to computations of Gr\"obner bases  
 in $D$ and primary ideal decompositions.  
 An implementation of this algorithm  
 uses sm1 for Gr\"obner basis computation in $D$ and  
 ox\_asir for primary ideal decomposition and Gr\"obner basis computation  
 in the ring of polynomials.  
   
 \subsubsection{A Course on Solving Algebraic Equations}  \subsubsection{A Course on Solving Algebraic Equations}
   
Line 73  which is good at Gr\"obner basis computations for zero
Line 71  which is good at Gr\"obner basis computations for zero
 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 used Risa/asir with ox\_sm1\_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\_sm1\_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
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