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version 1.2, 2000/01/02 07:32:11 version 1.4, 2000/01/15 01:33:32
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 % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.1 1999/12/23 10:25:08 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[page ??]{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}, which is a Gr\"obner engine for $D$,  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.
 For example, the following is the sm1 session to find the annihilating  These two OpenXM complient systems are integrated by
   OpenXM protocol.
   
   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}
 sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::  sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
Line 37  Byte order for engine process is network byte order.
Line 53  Byte order for engine process is network byte order.
      -7898796*s^3-5220720*s^2-1900500*s-294000]]       -7898796*s^3-5220720*s^2-1900500*s-294000]]
 \end{verbatim}  \end{verbatim}
 The last polynomial is factored as  The last polynomial is factored as
 $-12(s+1)(3s+5)(3s+4)(6*s+5)(6*s+7)$  $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
 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 used the text book \cite{CLO} which focuses  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.
 Risa/asir has a user language like C and we could teach a course  Risa/asir has a user language like C and we could teach a course
 with a unified environment  with a unified environment
 controlled by asir user language.  controlled by asir user language.
   The following is an asir session to solve algebraic equations by calling
   the PHC pack.
 \begin{verbatim}  \begin{verbatim}
 [257] phc([x^2+y^2-4,x*y-1]);  [257] phc([x^2+y^2-4,x*y-1]);
 The detailed output is in the file tmp.output.*  The detailed output is in the file tmp.output.*
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