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version 1.1, 2001/03/07 02:42:10 version 1.3, 2001/03/08 04:24:09
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 % $OpenXM$  % $OpenXM: OpenXM/doc/ascm2001/heterotic-network.tex,v 1.2 2001/03/07 06:54:40 takayama Exp $
 \section{Applications}  \section{Applications}
   
 \subsection{Heterogeneous Servers}  \subsection{Heterogeneous Servers}
Line 14  We can build a new computer math system by assembling
Line 14  We can build a new computer math system by assembling
 different OpenXM servers.  different OpenXM servers.
 It is similar to building a toy house by LEGO blocks.  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.  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}
Line 101  The answer is in the variable Phc.
Line 101  The answer is in the variable Phc.
 \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}  \caption{The first components of the solutions to the system of algebraic equations Katsura 7.}
 \label{katsura}  \label{katsura}
 \end{figure}  \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.
   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.
   
   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}
   
   
   
   
   
   
   
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