nn_ndbf

nn_ndbf User’s Manual

Edition 1.0

Nov 2009

by Masayuki Noro and Kenta Nishiyama

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

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

This function is defined in an asir-contrib package ‘nn_ndbf.rr’.
This function computes the global b-function of a polynomial f. By default only the global b-function is returned. If an option op=1 is given, a pair [b,P] of the global b-function and a differential operator satisfying Pf^(s+1)=b(s)f^s. The operator P is represented as a commutative polynomial of variables v1,...,vn,dv1,...,dvn. The d-variables are treated as commutative indeterminates in this representation and the polynomial should be regarded as a canonical representation with each polynomial coefficient placed at the left of d-variables.
If an option weight=[v1,w1,...,vn,wn] is given, the computation is done with a weight (w1,...,wn) for (v1,...,vn). This option is useful when f is weighted homogeneous with respect to (w1,...,wn).
If an option heuristic=1 is given a change of ordering is done before entering elimination. In some cases this improves the total efficiencty.
The variable order used in the whole computation is automatically set by default. If an option vord=v is given, a variable order v is used istead.

[...] 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

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

This function is defined in an asir-contrib package ‘nn_ndbf.rr’.
This function computes the local b-function of a polynomial f at a point (v1,...,vn)=(a1,...,an). The output is a list of pairs of each factor of the local b-function and its multiplicity.
By default only the local b-function is returned. If an option op=1 is given, a triple [b,a,P] of the local b-function, a polynomial and a differential operator satisfying Pf^(s+1)=ab(s)f^s. The operator P is represented as a commutative polynomial of variables v1,...,vn,dv1,...,dvn. The d-variables are treated as commutative indeterminates in this representation, the polynomial should be regarded as a canonical representation with each polynomial coefficient placed at the left of d-variables.
If an option weight=[v1,w1,...,vn,wn] is given, the computation is done with a weight (w1,...,wn) for (v1,...,vn). This option is useful when f is weighted homogeneous with respect to (w1,...,wn).
If an option heuristic=1 is given a change of ordering is done before entering elimination. In some cases this improves the total efficiencty.
The variable order used in the whole computation is automatically set by default. If an option vord=v is given, a variable order v is used istead.

[...] 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]

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

This function is defined in an asir-contrib package ‘nn_ndbf.rr’.
This function computes a stratification assoficated with local b-function of a polynomial f. The output is a list [s1,...sl] where each si is a list [l1,l2,bi]. In this list, l1 and l2 is generators of ideals and they represents the local b-function is bi over V(l1)-V(l2).
If an option weight=[v1,w1,...,vn,wn] is given, the computation is done with a weight (w1,...,wn) for (v1,...,vn). This option is useful when f is weighted homogeneous with respect to (w1,...,wn).
If an option heuristic=1 is given a change of ordering is done before entering elimination. In some cases this improves the total efficiencty.
The variable order used in the whole computation is automatically set by default. If an option vord=v is given, a variable order v is used istead.

[...] 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+...],[]]]

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])

This function computes the action of a differential operator op on gf^(s+a).
g is a polynomial with variables v1,...,vn.
op is represented by a polynonmial with [v1,...,vn,dv1,...,dvn].
The input list [g,f,s+a] represents gf^(s+a).
The result is a list [h,f,s+c] and it means hf^(s+c), where c is an integer. If op is an operator giving b-function b(s), then c=0 for a=1 and h=b(s) (global case) or h=b(s)d(v) (local case).

[...] 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]

0.2 Computation of annihilator ideal

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]

This function is defined in an asir-contrib package ‘nn_ndbf.rr’.
This function computes the annihilator ideal of f^s for f. The output is a list of defferential operators containing s in thier coefficients. The differential operators are represented in the same manner as ndbf.bf_local.
If an option weight=[v1,w1,...,vn,wn] is given, the computation is done with a weight (w1,...,wn) for (v1,...,vn). This option is useful when f is weighted homogeneous with respect to (w1,...,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]

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

0.1 Computation of b-function
0.2 Computation of annihilator ideal
- 0.2.1 ndbf.ann

Index

Short Table of Contents

Index

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