version 1.2, 2000/01/02 07:32:11 |
version 1.3, 2000/01/03 04:27:52 |
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% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.1 1999/12/23 10:25:08 takayama Exp $ |
% $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.2 2000/01/02 07:32:11 takayama Exp $ |
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\subsection{Heterotic Network} (Takayama) |
\subsection{Heterotic Network} (Takayama) |
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Line 17 Here, $\bullet$ denotes the action of $D$ to functions |
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Line 17 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[page ??]{sst-book}). |
was given by Oaku in 1995 (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}) |
An implementation of this algorithm (the function {\tt annfs}) |
on kan/sm1 \cite{kan}, which is a Gr\"obner engine for $D$, |
on kan/sm1 \cite{kan} |
calls ox\_asir to factorize polynomials to find the integral |
calls ox\_asir to factorize polynomials to find the integral |
roots. |
roots. |
For example, the following is the 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} |
sm1>[(x^3-y^2 z^2) (x,y,z)] annfs :: |
sm1>[(x^3-y^2 z^2) (x,y,z)] annfs :: |
Line 37 Byte order for engine process is network byte order. |
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Line 37 Byte order for engine process is network byte order. |
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-7898796*s^3-5220720*s^2-1900500*s-294000]] |
-7898796*s^3-5220720*s^2-1900500*s-294000]] |
\end{verbatim} |
\end{verbatim} |
The last polynomial is factored as |
The last polynomial is factored as |
$-12(s+1)(3s+5)(3s+4)(6*s+5)(6*s+7)$ |
$-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$ |
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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Line 77 We used Risa/asir with ox\_sm1\_phc (based on PHC pack |
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Line 77 We used Risa/asir with ox\_sm1\_phc (based on PHC pack |
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for the polyhedral homotopy method) and |
for the polyhedral homotopy method) and |
ox\_sm1\_gnuplot (GNUPLOT) servers |
ox\_sm1\_gnuplot (GNUPLOT) servers |
to teach a course on solving algebraic equations. |
to teach a course on solving algebraic equations. |
This course used the text book \cite{CLO} which focuses |
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 |
Risa/asir has a user language like C and we could teach a course |
with a unified environment |
with a unified environment |
controlled by asir user language. |
controlled by asir user language. |
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The following is an asir session to solve algebraic equations by calling |
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the PHC pack. |
\begin{verbatim} |
\begin{verbatim} |
[257] phc([x^2+y^2-4,x*y-1]); |
[257] phc([x^2+y^2-4,x*y-1]); |
The detailed output is in the file tmp.output.* |
The detailed output is in the file tmp.output.* |