version 1.3, 2000/01/03 04:27:52 |
version 1.5, 2000/01/15 03:23:59 |
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% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.2 2000/01/02 07:32:11 takayama Exp $ |
% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.4 2000/01/15 01:33:32 takayama Exp $ |
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\subsection{Heterotic Network} (Takayama) |
\subsection{Heterogeneous Servers} |
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\def\pd#1{ \partial_{#1} } |
\def\pd#1{ \partial_{#1} } |
\subsubsection{Annihilating ideal} |
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By using OpenXM, we can treat OpenXM servers essentially |
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like a subroutine. |
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Since OpenXM provides a universal stackmachine which does not |
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depend each servers, |
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it is relatively easy to install new servers. |
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We can build a new computer math system by assembling |
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different OpenXM servers. |
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It is similar to building a toy house by LEGO blocks. |
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We will see two examples of custom made systems |
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built by OpenXM servers. |
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\subsubsection{Computation of annihilating ideals by kan/sm1 and ox\_asir} |
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Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$ |
Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$ |
be the ring of differential operators. |
be the ring of differential operators. |
For a given polynomial |
For a given polynomial |
Line 17 Here, $\bullet$ denotes the action of $D$ to functions |
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Line 30 Here, $\bullet$ denotes the action of $D$ to functions |
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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[5.3]{sst-book}). |
was given by Oaku (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}) |
This algorithm (the function {\tt annfs}) is implemented by |
on kan/sm1 \cite{kan} |
kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and |
calls ox\_asir to factorize polynomials to find the integral |
ox\_asir, to factorize polynomials to find the integral |
roots. |
roots. |
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These two OpenXM complient systems are integrated by |
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OpenXM protocol. |
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For example, the following is a 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} |
Line 41 $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$ |
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Line 57 $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$ |
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and the minimal integral root is $-1$ |
and the minimal integral root is $-1$ |
as shown in the output. |
as shown in the output. |
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\subsubsection{Primary ideal decomposition and |
Similarly, |
a stratification of singularity} |
an algorithm to stratify singularity |
Let ${\cal D} = {\cal O}\langle \pd{1}, \ldots, \pd{n} \rangle$ |
\cite{oaku-advance} |
be the sheaf of algebraic differential operators on ${\bf C}^n$. |
is implemented by |
For a given polynomial |
kan/sm1 \cite{kan}, for Gr\"obner basis computation in $D$, and |
$f \in {\bf Q}[x_1, \ldots, x_n]$, |
ox\_asir, for primary ideal decompositions. |
minimal degree non-zero polynomial $b_a(s)$ such that |
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$ b_a(s) f^s \in {\cal D}_{a} \bullet f^{s+1} $ |
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is called the local $b$-function of $f^s$ at |
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$x=a$. |
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Here, ${\cal D}_a$ denotes the germ of the sheaf ${\cal D}$ at |
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$x=a$. |
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The polynomial $b_a(s)$ depends on $a$ and ${\bf C}^n$ is |
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stratified into a finite algebraic sets such that |
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$b_a(s)$ is a constant with respect to $a$ on each |
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algebraic set. |
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An algorithm to determine the stratification |
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was given by Oaku in 1997 \cite{oaku-advance}. |
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His algorithm reduces the problem to computations of Gr\"obner bases |
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in $D$ and primary ideal decompositions. |
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An implementation of this algorithm |
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uses sm1 for Gr\"obner basis computation in $D$ and |
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ox\_asir for primary ideal decomposition and Gr\"obner basis computation |
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in the ring of polynomials. |
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\subsubsection{A Course on Solving Algebraic Equations} |
\subsubsection{A Course on Solving Algebraic Equations} |
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Line 73 which is good at Gr\"obner basis computations for zero |
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Line 71 which is good at Gr\"obner basis computations for zero |
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with ${\bf Q}$ coefficients. |
with ${\bf Q}$ coefficients. |
However, it is not good at graphical presentations and |
However, it is not good at graphical presentations and |
numerical methods. |
numerical methods. |
We used Risa/asir with ox\_sm1\_phc (based on PHC pack by Verschelde \cite{phc} |
We integrated Risa/asir, ox\_phc (based on PHC pack by Verschelde \cite{phc} |
for the polyhedral homotopy method) and |
for the polyhedral homotopy method) and |
ox\_sm1\_gnuplot (GNUPLOT) servers |
ox\_gnuplot (GNUPLOT) servers |
to teach a course on solving algebraic equations. |
to teach a course on solving algebraic equations. |
This course was presented with the text book \cite{CLO} which discusses |
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 |
We could teach a course |
with a unified environment |
with a unified environment |
controlled by asir user language. |
controlled by asir user language, which is similar to C. |
The following is an asir session to solve algebraic equations by calling |
The following is an asir session to solve algebraic equations by calling |
the PHC pack. |
the PHC pack. |
\begin{verbatim} |
\begin{verbatim} |