Annotation of OpenXM/doc/issac2000/heterotic-network.tex, Revision 1.2
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:
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