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7.5.4.0. operatorBM
Procedure from library dmod.lib (see dmod_lib).
- Usage:
- operatorBM(f [,eng]); f a poly, eng an optional int
- Return:
- ring
- Purpose:
- compute the B-operator and other relevant data for Ann F^s,
using e.g. algorithm by Briancon and Maisonobe for Ann F^s and BS.
- Note:
- activate the output ring with the
setring command. In the output ring D[s]
- the polynomial F is the same as the input,
- the ideal LD is the annihilator of f^s in Dn[s],
- the ideal LD0 is the needed D-mod structure, where LD0 = LD|s=s0,
- the polynomial bs is the global Bernstein polynomial of f in the variable s,
- the list BS contains all the roots with multiplicities of the global Bernstein polynomial of f,
- the polynomial PS is an operator in Dn[s] such that PS*f^(s+1) = bs*f^s.
If eng <>0, std is used for Groebner basis computations,
otherwise and by default slimgb is used.
- Display:
- If
printlevel=1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.
Example:
| | LIB "dmod.lib";
ring r = 0,(x,y,z),Dp;
poly F = x^3+y^3+z^3;
printlevel = 0;
def A = operatorBM(F);
setring A;
F; // the original polynomial itself
==> x^3+y^3+z^3
LD; // generic annihilator
==> LD[1]=x*Dx+y*Dy+z*Dz-3*s
==> LD[2]=z^2*Dy-y^2*Dz
==> LD[3]=z^2*Dx-x^2*Dz
==> LD[4]=y^2*Dx-x^2*Dy
==> LD[5]=x^3*Dz+y^3*Dz+z^3*Dz-3*z^2*s
==> LD[6]=x^3*Dy+y^3*Dy+y^2*z*Dz-3*y^2*s
LD0; // annihilator
==> LD0[1]=x*Dx+y*Dy+z*Dz+6
==> LD0[2]=z^2*Dy-y^2*Dz
==> LD0[3]=z^2*Dx-x^2*Dz
==> LD0[4]=y^2*Dx-x^2*Dy
==> LD0[5]=x^3*Dz+y^3*Dz+z^3*Dz+6*z^2
==> LD0[6]=x^3*Dy+y^3*Dy+y^2*z*Dz+6*y^2
bs; // normalized Bernstein poly
==> s^5+7*s^4+173/9*s^3+233/9*s^2+154/9*s+40/9
BS; // roots and multiplicities of the Bernstein poly
==> [1]:
==> _[1]=-1
==> _[2]=-5/3
==> _[3]=-4/3
==> _[4]=-2
==> [2]:
==> 2,1,1,1
PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD
==> 2/81*y*z*Dx^3*Dy*Dz-2/81*y*z*Dy^4*Dz-4/81*y^2*Dy^2*Dz^3-2/81*y*z*Dy*Dz^4+\
2/81*y*Dx^3*Dy*s-2/81*y*Dy^4*s+2/81*z*Dx^3*Dz*s+2/27*z*Dy^3*Dz*s+2/27*y*D\
y*Dz^3*s-2/81*z*Dz^4*s+2/27*y*Dx^3*Dy-2/27*y*Dy^4+2/27*z*Dx^3*Dz+2/27*z*D\
y^3*Dz-10/81*y*Dy*Dz^3-2/27*z*Dz^4+1/27*Dx^3*s^2+1/9*Dy^3*s^2+1/9*Dz^3*s^\
2+5/27*Dx^3*s+11/27*Dy^3*s+11/27*Dz^3*s+20/81*Dx^3+8/27*Dy^3+16/81*Dz^3
reduce(PS*F-bs,LD); // check the property of PS
==> 0
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