bfunction, bfct, generic_bfct, ann, ann0

bfunction(f)

bfct(f)

generic_bfct(plist,vlist,dvlist,weight)

:: Computes the global b function of a polynomial or an ideal

ann(f)

ann0(f)

:: Computes the annihilator of a power of polynomial

return

polynomial or list

f

polynomial

plist

list of polynomials

vlist dvlist

list of variables

• These functions are defined in `bfct'.
• bfunction(f) and bfct(f) compute the global b-function b(s) of a polynomial f. b(s) is a polynomial of the minimal degree such that there exists P(x,s) in D[s], which is a polynomial ring over Weyl algebra D, and P(x,s)f^(s+1)=b(s)f^s holds.
• generic_bfct(f,vlist,dvlist,weight) computes the global b-function of a left ideal I in D generated by plist, with respect to weight. vlist is the list of x-variables, vlist is the list of corresponding D-variables.
• bfunction(f) and bfct(f) implement different algorithms and the efficiency depends on inputs.
• ann(f) returns the generator set of the annihilator ideal of f^s. ann(f) returns a list [a,list], where a is the minimal integral root of the global b-function of f, and list is a list of polynomials obtained by substituting s in ann(f) with a.
• See [Saito,Sturmfels,Takayama] for the details.

[0] load("bfct")$
[216] bfunction(x^3+y^3+z^3+x^2*y^2*z^2+x*y*z);
-9*s^5-63*s^4-173*s^3-233*s^2-154*s-40
[217] fctr(@);
[[-1,1],[s+2,1],[3*s+4,1],[3*s+5,1],[s+1,2]]
[218] F = [4*x^3*dt+y*z*dt+dx,x*z*dt+4*y^3*dt+dy,
x*y*dt+5*z^4*dt+dz,-x^4-z*y*x-y^4-z^5+t]$
[219] generic_bfct(F,[t,z,y,x],[dt,dz,dy,dx],[1,0,0,0]);
20000*s^10-70000*s^9+101750*s^8-79375*s^7+35768*s^6-9277*s^5
+1278*s^4-72*s^3
[220] P=x^3-y^2$
[221] ann(P);
[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y+6*s]
[222] ann0(P);
[-1,[2*dy*x+3*dx*y^2,-3*dx*x-2*dy*y-6]]

References

section Weyl algebra.

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