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Annotation of OpenXM/doc/issac2000/heterotic-network.tex, Revision 1.3

1.3     ! takayama    1: % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.2 2000/01/02 07:32:11 takayama Exp $
1.2       takayama    2:
                      3: \subsection{Heterotic Network}   (Takayama)
                      4:
                      5: \def\pd#1{ \partial_{#1} }
                      6: \subsubsection{Annihilating ideal}
                      7:
                      8: Let $D = {\bf Q} \langle x_1, \ldots, x_n , \pd{1}, \ldots, \pd{n} \rangle$
                      9: be the ring of differential operators.
                     10: For a given polynomial
                     11: $ f \in {\bf Q}[x_1, \ldots, x_n] $,
                     12: the annihilating ideal of $f^{-1}$ is defined as
                     13: $$ {\rm Ann}\, f^{-1} = \{ \ell \in D \,|\,
                     14:   \ell \bullet f^{-1} = 0 \}.
                     15: $$
                     16: Here, $\bullet$ denotes the action of $D$ to functions.
                     17: The annihilating ideal can be regarded as the maximal differential
                     18: equations for the function $f^{-1}$.
                     19: An algorithm to determine generators of the annihilating ideal
1.3     ! takayama   20: was given by Oaku in 1995 (see, e.g., \cite[5.3]{sst-book}).
1.2       takayama   21: His algorithm reduces the problem to computations of Gr\"obner bases
                     22: in $D$ and to find the maximal integral root of a polynomial.
                     23: An implementation of this algorithm (the function {\tt annfs})
1.3     ! takayama   24: on kan/sm1 \cite{kan}
1.2       takayama   25: calls ox\_asir to factorize polynomials to find the integral
                     26: roots.
1.3     ! takayama   27: For example, the following is a sm1 session to find the annihilating
1.2       takayama   28: ideal for $f = x^3 - y^2 z^2$.
                     29: \begin{verbatim}
                     30: sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
                     31: Starting ox_asir server.
                     32: Byte order for control process is network byte order.
                     33: Byte order for engine process is network byte order.
                     34: [[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
                     35:   -2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx],
                     36:  [-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
                     37:      -7898796*s^3-5220720*s^2-1900500*s-294000]]
                     38: \end{verbatim}
                     39: The last polynomial is factored as
1.3     ! takayama   40: $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$
1.2       takayama   41: and the minimal integral root is $-1$
                     42: as shown in the output.
                     43:
                     44: \subsubsection{Primary ideal decomposition and
                     45: a stratification of singularity}
                     46: Let ${\cal D} = {\cal O}\langle \pd{1}, \ldots, \pd{n} \rangle$
                     47: be the sheaf of algebraic differential operators on ${\bf C}^n$.
                     48: For a given polynomial
                     49: $f \in {\bf Q}[x_1, \ldots, x_n]$,
                     50: minimal degree non-zero polynomial $b_a(s)$ such that
                     51: $ b_a(s) f^s \in {\cal D}_{a} \bullet f^{s+1} $
                     52: is called the local $b$-function of $f^s$ at
                     53: $x=a$.
                     54: Here, ${\cal D}_a$ denotes the germ of the sheaf ${\cal D}$ at
                     55: $x=a$.
                     56: The polynomial $b_a(s)$ depends on $a$ and ${\bf C}^n$ is
                     57: stratified into a finite algebraic sets such that
                     58: $b_a(s)$ is a constant with respect to $a$ on each
                     59: algebraic set.
                     60: An algorithm to determine the stratification
                     61: was given by Oaku in 1997 \cite{oaku-advance}.
                     62: His algorithm reduces the problem to computations of Gr\"obner bases
                     63: in $D$ and primary ideal decompositions.
                     64: An implementation of this algorithm
                     65: uses sm1 for Gr\"obner basis computation in $D$ and
                     66: ox\_asir for primary ideal decomposition and Gr\"obner basis computation
                     67: in the ring of polynomials.
                     68:
                     69: \subsubsection{A Course on Solving Algebraic Equations}
                     70:
                     71: Risa/asir \cite{asir} is a general computer algebra system
                     72: which is good at Gr\"obner basis computations for zero dimensional ideal
                     73: with ${\bf Q}$ coefficients.
                     74: However, it is not good at graphical presentations and
                     75: numerical methods.
                     76: We used Risa/asir with ox\_sm1\_phc (based on PHC pack by Verschelde \cite{phc}
                     77: for the polyhedral homotopy method) and
                     78: ox\_sm1\_gnuplot (GNUPLOT) servers
                     79: to teach a course on solving algebraic equations.
1.3     ! takayama   80: This course was presented with the text book \cite{CLO} which discusses
1.2       takayama   81: on the Gr\"obner basis method and the polyhedral homotopy method
                     82: to solve systems of algebraic equations.
                     83: Risa/asir has a user language like C and we could teach a course
                     84: with a unified environment
                     85: controlled by asir user language.
1.3     ! takayama   86: The following is an asir session to solve algebraic equations by calling
        !            87: the PHC pack.
1.2       takayama   88: \begin{verbatim}
                     89: [257] phc([x^2+y^2-4,x*y-1]);
                     90: The detailed output is in the file tmp.output.*
                     91: The answer is in the variable Phc.
                     92: 0
                     93: [260] Phc ;
                     94: [[[-0.517638,0],[-1.93185,0]],
                     95: [[1.93185,0],[0.517638,0]],
                     96: [[-1.93185,0],[-0.517638,0]],
                     97: [[0.517638,0],[1.93185,0]]]
                     98: [261]
                     99: \end{verbatim}
                    100:
                    101:
                    102: