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:
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