nn_ndbf

nn_ndbf User’s Manual

Edition 1.0

Nov 2009

by Masayuki Noro and Kenta Nishiyama

Copyright © Masayuki Noro and Kenta Nishiyama 2009. All rights reserved.


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In this manual we explain about a new b-function package ‘nn_ndbf.rr’ in asir-contrib. To use this package one has to load ‘nn_ndbf.rr’.

[...] load("nn_ndbf.rr");

A prefix ndbf. is necessary to call the functions in this package. In this manual we also explain about some related built-in functions.

0.1 Computation of b-function


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0.1.1 ndbf.bfunction

ndbf.bfunction(f[|weight=w,heruristic=yesno,vord=v,op=yesno]) :: computes the global b-function of a polynomial f
return

a polynomial

f

a polynomial

w

a list [v1,w1,...,vn,wn]

yesno

0 or 1

v

a list of variables

[...] load("nn_ndbf.rr");
[...] ndbf.bfunction(x^3-y^2*z^2);
-11664*s^7-93312*s^6-316872*s^5-592272*s^4-658233*s^3-435060*s^2
-158375*s-24500
[...] ndbf.bfunction(x^3-y^2*z^2|op=1);
[-11664*s^7-93312*s^6-316872*s^5-592272*s^4-658233*s^3-435060*s^2
-158375*s-24500,(108*z^3*x*dz^3+756*z^2*x*dz^2+1080*z*x*dz+216*x)*dx^4
...
+(729/8*z^3*dz^5+9477/8*z^2*dz^4+5103/2*z*dz^3+2025/2*dz^2)*dy^2]
[...] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4
-4*u3^3*u2^2$
[...] ndbf.bfunction(F|weight=[u3,2,u2,3,u1,4]);
576*s^6+3456*s^5+8588*s^4+11312*s^3+8329*s^2+3250*s+525

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0.1.2 ndbf.bf_local

ndbf.bf_local(f,p[|weight=w,heruristic=yesno,vord=v,op=yesno]) :: computes the local b-function of a polynomial f at p.
return

a list

f

a polynomail

p

a list [v1,a1,...,vn,an]

w

a list [v1,w1,...,vn,wn]

yesno

0 or 1

v

a list of variables

[...] load("nn_ndbf.rr");
[...] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]);
[[-s-1,2]]
[...] ndbf.bf_local(y*((x+1)*x^3-y^2),[x,-1,y,0]|op=1);
[[[-s-1,2]],12*x^3+36*y^2*x-36*y^2,(32*y*x^2+56*y*x)*dx^2
+((-8*x^3-2*x^2+(128*y^2-6)*x+112*y^2)*dy+288*y*x+(-240*s-128)*y)*dx
+(32*y*x^2-6*y*x+128*y^3-9*y)*dy^2+(32*x^2+6*s*x+640*y^2+39*s+30)*dy
+(-1152*s^2-3840*s-2688)*y]

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0.1.3 ndbf.bf_strat

ndbf.bf_strat(f[|weight=w,heruristic=h,vord=v])

:: computes a stratification associated with local b-function of a polynomial f.

return

a list

f

a polynomial

w

a list [v1,w1,...,vn,wn]

h

0 or 1

v

li ist of variables

[...] load("nn_ndbf.rr");
[...] F=256*u1^3-128*u3^2*u1^2+(144*u3*u2^2+16*u3^4)*u1-27*u2^4
-4*u3^3*u2^2$
[...] ndbf.bf_strat(F);
[[[u3^2,-u1,-u2],[-1],[[-s-1,2],[16*s^2+32*s+15,1],[36*s^2+72*s+35,1]]],
[[-4*u1+u3^2,-u2],[96*u1^2+40*u3^2*u1-9*u3*u2^2,...],[[-s-1,2]]],
[[-2048*u1^3-...],[-u3*u2,u2*u1,...],[[-s-1,1],...]]],
[[-256*u1^3+128*u3^2*u1^2+...],[...],[[-s-1,1]]],
[[],[-256*u1^3+128*u3^2*u1^2+...],[]]]

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0.1.4 ndbf.action_on_gfs

ndbf.action_on_gfs(op,v,gfs)

:: computes the action of an operatior op on gf^(s+a)

return

a list

op

a differential operator

gfs

a list [g,f,s+a]

v

list of variables of f (v=[v1,...,vn])

[...] load("nn_ndbf.rr");
[...] F=x^5-y^2*z^2$
[...] B=ndbf.bfunction(F|op=1)$
[...] ndbf.action_on_gfs(B[1],[x,y,z],[1,F,s+1]);
[-62500000000*s^13-...-2985505717194*s-245434132944,x^5-z^2*y^2,s]
[...] L=ndbf.bf_local(F,[x,0,y,0,z,1]|op=1)$     
[...] ndbf.action_on_gfs(L[2],[x,y,z],[1,F,s+1]);
[(-100000*s^5-500000*s^4-990000*s^3-970000*s^2-470090*s-90090)*z^2,
x^5-z^2*y^2,s]

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0.2 Computation of annihilator ideal


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0.2.1 ndbf.ann

ndbf.ann(f[|weight=w]) :: computes the annihilator ideal of f^s for a polynomial f.
return

a list of differential operators

f

a polynomial

w

a list [v0,w1,...,vn,wn]

[...] load("nn_ndbf.rr");
[...] ndbf.ann(x*y*z*(x^3-y^2*z^2));
[(-x^4*dy^2+3*z^4*x*dz^2+12*z^3*x*dz+6*z^2*x)*dx+4*z*x^3*dz*dy^2
-z^5*dz^3-6*z^4*dz^2-6*z^3*dz,
(x^4*dy-3*z^3*y*x*dz-6*z^2*y*x)*dx-4*z*x^3*dz*dy+z^4*y*dz^2+3*z^3*y*dz,
(-x^4+3*z^2*y^2*x)*dx+(4*z*x^3-z^3*y^2)*dz,2*x*dx+3*z*dz-11*s,
-y*dy+z*dz]

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Index

Jump to:   N  
Index Entry  Section

N
ndbf.action_on_gfs 0.1.4 ndbf.action_on_gfs
ndbf.ann 0.2.1 ndbf.ann
ndbf.bfunction 0.1.1 ndbf.bfunction
ndbf.bf_local 0.1.2 ndbf.bf_local
ndbf.bf_strat 0.1.3 ndbf.bf_strat

Jump to:   N  

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