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-% $OpenXM$
+% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.1 1999/12/23 10:25:08 takayama Exp $
+
+\subsection{Heterotic Network}   (Takayama)
+
+\def\pd#1{ \partial_{#1} }
+\subsubsection{Annihilating ideal}
+
+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 in 1995 (see, e.g., \cite[page ??]{sst-book}).
+His algorithm reduces the problem to computations of Gr\"obner bases
+in $D$ and to find the maximal integral root of a polynomial.
+An implementation of this algorithm (the function {\tt annfs})
+on kan/sm1 \cite{kan}, which is a Gr\"obner engine for $D$,
+calls ox\_asir to factorize polynomials to find the integral
+roots.
+For example, the following is the sm1 session to find the annihilating
+ideal for $f = x^3 - y^2 z^2$.
+\begin{verbatim}
+sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
+Starting ox_asir server.
+Byte order for control process is network byte order.
+Byte order for engine process is network byte order.
+[[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy, 
+  -2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx], 
+ [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
+     -7898796*s^3-5220720*s^2-1900500*s-294000]] 
+\end{verbatim}
+The last polynomial is factored as
+$-12(s+1)(3s+5)(3s+4)(6*s+5)(6*s+7)$
+and the minimal integral root is $-1$
+as shown in the output.
+
+\subsubsection{Primary ideal decomposition and
+a stratification of singularity}
+Let ${\cal D} = {\cal O}\langle \pd{1}, \ldots, \pd{n} \rangle$
+be the sheaf of algebraic differential operators on ${\bf C}^n$.
+For a given polynomial
+$f \in {\bf Q}[x_1, \ldots, x_n]$,
+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}
+
+Risa/asir \cite{asir} is a general computer algebra system
+which is good at Gr\"obner basis computations for zero dimensional ideal
+with ${\bf Q}$ coefficients.
+However, it is not good at graphical presentations and
+numerical methods.
+We used Risa/asir with ox\_sm1\_phc (based on PHC pack by Verschelde \cite{phc}
+for the polyhedral homotopy method) and
+ox\_sm1\_gnuplot (GNUPLOT) servers
+to teach a course on solving algebraic equations.
+This course used the text book \cite{CLO} which focuses
+on the Gr\"obner basis method and the polyhedral homotopy method
+to solve systems of algebraic equations.
+Risa/asir has a user language like C and we could teach a course
+with a unified environment
+controlled by asir user language.
+\begin{verbatim}
+[257] phc([x^2+y^2-4,x*y-1]);
+The detailed output is in the file tmp.output.*
+The answer is in the variable Phc.
+0
+[260] Phc ;
+[[[-0.517638,0],[-1.93185,0]],
+[[1.93185,0],[0.517638,0]],
+[[-1.93185,0],[-0.517638,0]],
+[[0.517638,0],[1.93185,0]]]
+[261] 
+\end{verbatim}
+
+
+