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version 1.2, 2000/01/02 07:32:11 version 1.3, 2000/01/03 04:27:52
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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 1.2 2000/01/02 07:32:11 takayama Exp $
   
 \subsection{Heterotic Network}   (Takayama)  \subsection{Heterotic Network}   (Takayama)
   
Line 17  Here, $\bullet$ denotes the action of $D$ to functions
Line 17  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 in 1995 (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})  An implementation of this algorithm (the function {\tt annfs})
 on kan/sm1 \cite{kan}, which is a Gr\"obner engine for $D$,  on kan/sm1 \cite{kan}
 calls ox\_asir to factorize polynomials to find the integral  calls ox\_asir to factorize polynomials to find the integral
 roots.  roots.
 For example, the following is the 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}
 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 37  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.
   
Line 77  We used Risa/asir with ox\_sm1\_phc (based on PHC pack
Line 77  We used Risa/asir with ox\_sm1\_phc (based on PHC pack
 for the polyhedral homotopy method) and  for the polyhedral homotopy method) and
 ox\_sm1\_gnuplot (GNUPLOT) servers  ox\_sm1\_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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