[BACK]Return to heterotic-network.tex CVS log [TXT][DIR] Up to [local] / OpenXM / doc / issac2000

Diff for /OpenXM/doc/issac2000/heterotic-network.tex between version 1.1 and 1.2

version 1.1, 1999/12/23 10:25:08 version 1.2, 2000/01/02 07:32:11
Line 1 
Line 1 
 % $OpenXM$  % $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}
   
   
   
Legend:
Removed from v.1.1  
changed lines
  Added in v.1.2
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>