===================================================================
RCS file: /home/cvs/OpenXM/doc/ascm2001/heterotic-network.tex,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -p -r1.1 -r1.2
--- OpenXM/doc/ascm2001/heterotic-network.tex	2001/03/07 02:42:10	1.1
+++ OpenXM/doc/ascm2001/heterotic-network.tex	2001/03/07 06:54:40	1.2
@@ -1,4 +1,4 @@
-% $OpenXM$
+% $OpenXM: OpenXM/doc/ascm2001/heterotic-network.tex,v 1.1 2001/03/07 02:42:10 noro Exp $
 \section{Applications}
 
 \subsection{Heterogeneous Servers}
@@ -14,7 +14,7 @@ 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 custom-made systems
+We will see three examples of custom-made systems
 built by OpenXM servers.
 
 \subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir}
@@ -101,6 +101,99 @@ The answer is in the variable Phc.
 \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
 \label{katsura}
 \end{figure}
+
+\subsubsection{Asir-contrib-HG package to solve GKZ hypergeometric systems}
+
+GKZ hypergeometric system is a system of linear partial differential
+equations associated to $A=(a_{ij})$  
+(an integer $d\times n$-matrix of rank $d$)
+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
+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,
+and
+(2) enumeration of all the Gr\"obner bases of toric ideals.
+{\tt Asir} and {\tt Kan/sm1} provide functions for (1) and
+{\tt TiGERS} provides a function for (2).
+These components communicate each other by OpenXM-RFC 100 protocol.
+
+Let us see an example how to construct series solution of a GKZ hypergeometric
+system.
+The function
+{\tt dsolv\_starting\_term} finds the leading terms of series solutions
+to a given direction.
+\begin{enumerate}
+\item Generate the GKZ hypergeometric system associated to
+                    $\pmatrix{ 1&1&1&1&1 \cr
+                              1&1&0&-1&0 \cr
+                              0&1&1&-1&0 \cr}$
+by the function {\tt sm1\_gkz}.
+\begin{verbatim}
+[1076] F = sm1_gkz( 
+      [ [[1,1,1,1,1],
+         [1,1,0,-1,0],
+         [0,1,1,-1,0]], [1,0,0]]);
+[[x5*dx5+x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,
+  -x4*dx4+x2*dx2+x1*dx1,
+  -x4*dx4+x3*dx3+x2*dx2,
+  -dx2*dx5+dx1*dx3,dx5^2-dx2*dx4],
+ [x1,x2,x3,x4,x5]]
+\end{verbatim}
+\item Find the leading terms of this system to the direction
+$(1,1,1,1,0)$.
+\begin{verbatim}
+[1077] A= dsolv_starting_term(F[0],F[1],
+                            [1,1,1,1,0])$
+Computing the initial ideal.
+Done.
+Computing a primary ideal decomposition.
+Primary ideal decomposition of 
+the initial Frobenius ideal
+to the direction [1,1,1,1,0] is
+[[[x5+2*x4+x3-1,x5+3*x4-x2-1,
+   x5+2*x4+x1-1,3*x5^2+(8*x4-6)*x5-8*x4+3,
+   x5^2-2*x5-8*x4^2+1,x5^3-3*x5^2+3*x5-1],
+ [x5-1,x4,x3,x2,x1]]]
+
+----------- root is [ 0 0 0 0 1 ]
+----------- dual system is
+[x5^2+(-3/4*x4-1/2*x3-1/4*x2-1/2*x1)*x5+1/8*x4^2
+ +(1/4*x3+1/4*x1)*x4+1/4*x2*x3-1/8*x2^2+1/4*x1*x2,
+ x4-2*x3+3*x2-2*x1,x5-x3+x2-x1,1]
+\end{verbatim}
+\item From the output, we can see that we have four possible 
+leading terms.
+Factoring these leading terms, we get the following simpler expressions.
+The third entry
+{\tt [[1,1],[x5,1],[-log(x1)+log(x2)-log(x3)+log(x5),1]], }
+means that there exists a series solution which starts with
+\[
+x_5 (-\log x_1 + \log x_2 - \log x_3 + \log x_5) =
+   x_5 \log \frac{x_2 x_5}{x_1 x_3}
+\]
+\begin{verbatim}
+[1078] A[0];
+[[ 0 0 0 0 1 ]]
+[1079] map(fctr,A[1][0]);
+[[[1/8,1],[x5,1],[log(x2)+log(x4)-2*log(x5),1],
+   [2*log(x1)-log(x2)+2*log(x3)+log(x4)-4*log(x5)
+    ,1]],
+ [[1,1],[x5,1],
+   [-2*log(x1)+3*log(x2)-2*log(x3)+log(x4),1]],
+ [[1,1],[x5,1],
+   [-log(x1)+log(x2)-log(x3)+log(x5),1]],
+ [[1,1],[x5,1]]]
+\end{verbatim}
+\end{enumerate}
+
+
+
+