===================================================================
RCS file: /home/cvs/OpenXM/doc/ascm2001p/heterotic-network.tex,v
retrieving revision 1.1
retrieving revision 1.6
diff -u -p -r1.1 -r1.6
--- OpenXM/doc/ascm2001p/heterotic-network.tex	2001/06/19 07:32:58	1.1
+++ OpenXM/doc/ascm2001p/heterotic-network.tex	2001/06/21 03:13:35	1.6
@@ -1,4 +1,4 @@
-% $OpenXM$
+% $OpenXM: OpenXM/doc/ascm2001p/heterotic-network.tex,v 1.5 2001/06/21 00:15:34 takayama Exp $
 \section{Applications}
 
 \subsection{Heterogeneous Servers}
@@ -8,39 +8,41 @@
 By using OpenXM, we can treat OpenXM servers essentially 
 like a subroutine.
 Since OpenXM provides a universal stack machine which does not
-depend each servers, 
+depend on each server, 
 it is relatively easy to install new servers.
-We can build a new computer math system by assembling
+We can build a new mathematical software system by assembling
 different OpenXM servers.
-It is similar to building a toy house by LEGO blocks.
+OpenXM servers currently provide 1077 functions
+\cite{openxm-1077}.
+We can use these as building blocks for a new system.
 
-We will see two examples of custom-made systems
-built by OpenXM servers.
+%We present an example of a custom-made system
+%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}
+%
+%Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
+%be the ring of differential operators.
+%For a given polynomial
+%$ f \in {\bf Q}[x_1, \ldots, x_n] $,
+%the annihilating ideal of $f^{-1}$ is defined as
+%$$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
+%  \ell \bullet f^{-1} = 0 \}.
+%$$
+%Here, $\bullet$ denotes the action of $D$ to functions.
+%The annihilating ideal can be regarded as the maximal differential
+%equations for the function $f^{-1}$.
+%An algorithm to determine generators of the annihilating ideal
+%was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
+%His algorithm reduces the problem to computations of Gr\"obner bases
+%in $D$ and to find the minimal integral root of a polynomial.
+%This algorithm (the function {\tt annfs}) is implemented by
+%kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
+%{\tt ox\_asir}, to factorize polynomials to find the integral
+%roots.
+%These two OpenXM compliant systems are integrated by
+%the OpenXM protocol.
 
-Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
-be the ring of differential operators.
-For a given polynomial
-$ f \in {\bf Q}[x_1, \ldots, x_n] $,
-the annihilating ideal of $f^{-1}$ is defined as
-$$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
-  \ell \bullet f^{-1} = 0 \}.
-$$
-Here, $\bullet$ denotes the action of $D$ to functions.
-The annihilating ideal can be regarded as the maximal differential
-equations for the function $f^{-1}$.
-An algorithm to determine generators of the annihilating ideal
-was given by Oaku (see, e.g., \cite[5.3]{sst-book}).
-His algorithm reduces the problem to computations of Gr\"obner bases
-in $D$ and to find the minimal integral root of a polynomial.
-This algorithm (the function {\tt annfs}) is implemented by
-kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and
-{\tt ox\_asir}, to factorize polynomials to find the integral
-roots.
-These two OpenXM compliant systems are integrated by
-the OpenXM protocol.
-
 %For example, the following is a sm1 session to find the annihilating
 %ideal for $f = x^3 - y^2 z^2$.
 %\begin{verbatim}
@@ -102,8 +104,8 @@ the OpenXM protocol.
 %\label{katsura}
 %\end{figure}
 
-\subsubsection{Asir-contrib-HG package to solve GKZ hypergeometric systems}
-
+Asir-contrib/Hypergeometric package is an 
+example of a custom-made system built by OpenXM servers.
 GKZ hypergeometric system is a system of linear partial differential
 equations associated to $A=(a_{ij})$  
 (an integer $d\times n$-matrix of rank $d$)
@@ -111,16 +113,23 @@ and $\beta \in {\bf C}^d$.
 The book by Saito, Sturmfels and Takayama \cite{sst-book}
 discusses algorithmic methods to construct series solutions of the GKZ
 system.
-The current Asir-contrib-HG package is built in order to implement
+The current Asir-contrib/Hypergeoemtric package is built in order to implement
 these algorithms.
 What we need for the implementation are mainly
 (1) Gr\"obner basis computation both in the ring of polynomials
 and in the ring of differential operators,
+(2) enumeration of all the Gr\"obner bases of toric ideals,
+and 
+(3) primary ideal decomposition.
+Asir and kan/sm1 provide functions for (1),
+{\tt TiGERS} provides a function for (2),
 and
-(2) enumeration of all the Gr\"obner bases of toric ideals.
-Asir and kan/sm1 provide functions for (1) and
-{\tt TiGERS} provides a function for (2).
-These components communicate each other by OpenXM-RFC 100 protocol.
+Asir provides a function for (3).
+These software systems communicate with each other 
+by the OpenXM-RFC 100 protocol
+and form a unified package for GKZ hypergeometric systems.
+See the chapter of {\tt dsolv} of Asir Contrib User's manual \cite{openxm-web}
+for details.
 
 %Let us see an example how to construct series solution of a GKZ hypergeometric
 %system.