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1.2     ! takayama    1: % $OpenXM: OpenXM/doc/issac2000/heterotic-network.tex,v 1.1 1999/12/23 10:25:08 takayama Exp $
        !             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
        !            20: was given by Oaku in 1995 (see, e.g., \cite[page ??]{sst-book}).
        !            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})
        !            24: on kan/sm1 \cite{kan}, which is a Gr\"obner engine for $D$,
        !            25: calls ox\_asir to factorize polynomials to find the integral
        !            26: roots.
        !            27: For example, the following is the sm1 session to find the annihilating
        !            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
        !            40: $-12(s+1)(3s+5)(3s+4)(6*s+5)(6*s+7)$
        !            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.
        !            80: This course used the text book \cite{CLO} which focuses
        !            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.
        !            86: \begin{verbatim}
        !            87: [257] phc([x^2+y^2-4,x*y-1]);
        !            88: The detailed output is in the file tmp.output.*
        !            89: The answer is in the variable Phc.
        !            90: 0
        !            91: [260] Phc ;
        !            92: [[[-0.517638,0],[-1.93185,0]],
        !            93: [[1.93185,0],[0.517638,0]],
        !            94: [[-1.93185,0],[-0.517638,0]],
        !            95: [[0.517638,0],[1.93185,0]]]
        !            96: [261]
        !            97: \end{verbatim}
        !            98:
        !            99:
        !           100: