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7.5.25.0. ratstd
Procedure from library ratgb.lib (see ratgb_lib).
- Usage:
- ratstd(I, n [,eng]); I an ideal/module, n an integer, eng an optional integer
- Return:
- ring
- Purpose:
- compute the Groebner basis of I in the Ore localization of
the basering with respect to the subalgebra, generated by first n variables
- Assume:
- the variables of basering are organized in two blocks and
- the first block of length
n contains the elements with respect to which one localizes,
- the basering is equipped with anti-block ordering, giving block
dominance for the variables in the second block
- Note:
- the output ring C is commutative. The ideal
rGBid in C
represents the rational form of the output ideal pGBid in the basering.
- During the computation, the D-dimension of I, Ratgb::Ddim and the corresponding
dimension as K(x)-vector space of I (Ratgb::KXdim, if Ratgb::Ddim=0) are computed and exported.
- Setting optional integer eng to 1, std is taken as Groebner engine; default is slimgb.
- Display:
- In order to see the steps of the computation, set printlevel to >=2
Example:
| | LIB "ratgb.lib";
ring r = (0,c),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
// this ordering is an antiblock ordering, as it must be
def S = Weyl(); setring S;
// the ideal I below annihilates parametric Appel F4 function
// where we set parameters to a=-2, b=-1 and d=0
ideal I =
x*Dx*(x*Dx+c-1) - x*(x*Dx+y*Dy-2)*(x*Dx+y*Dy-1),
y*Dy*(y*Dy-1) - y*(x*Dx+y*Dy-2)*(x*Dx+y*Dy-1);
int is = 2; // hence 1st and 2nd variables, that is x and y
// will become invertible in the localization
def A = ratstd(I,2); // main call
pGBid; // polynomial form of the basis in the localized ring
==> pGBid[1]=2*x*y*Dx*Dy+x*y*Dy^2+y^2*Dy^2-y*Dy^2+(-c-2)*x*Dx-2*y*Dy+2
==> pGBid[2]=x*Dx^2-y*Dy^2+(c)*Dx
==> pGBid[3]=2*x^2*y*Dy^3-4*x*y^2*Dy^3+2*y^3*Dy^3-4*x*y*Dy^3-4*y^2*Dy^3+2*y*D\
y^3+(-2*c)*x^2*Dx*Dy+(2*c)*x*Dx*Dy+2*x^2*Dy^2+(c-2)*x*y*Dy^2+(-3*c)*y^2*D\
y^2-4*x*Dy^2+(3*c-2)*y*Dy^2+2*Dy^2+(-c^2+2*c)*x*Dx+(6*c)*y*Dy+(-6*c)
setring A; // A is a commutative ring used for presentation
rGBid; // "rational" or "localized" form of the basis
==> rGBid[1]=(2*x*y)*Dx*Dy+(x*y+y^2-y)*Dy^2+(-c*x-2*x)*Dx+(-2*y)*Dy+2
==> rGBid[2]=(x)*Dx^2+(-y)*Dy^2+(c)*Dx
==> rGBid[3]=(2*x^2*y-4*x*y^2-4*x*y+2*y^3-4*y^2+2*y)*Dy^3+(-2*c*x^2+2*c*x)*Dx\
*Dy+(c*x*y-3*c*y^2+3*c*y+2*x^2-2*x*y-4*x-2*y+2)*Dy^2+(-c^2*x+2*c*x)*Dx+(6\
*c*y)*Dy+(-6*c)
Ratgb::Ddim; // the Krull-like dimension of A/I
==> 0
Ratgb::KXdim; // the dimension of A/I as a left K(x,y)-vector space
==> 4
//--- Now, let us compute a K(x,y) basis explicitly
print(matrix(kbase(rGBid)));
==> // ** rGBid is no standard basis
==> Dy^2,Dy,Dx,1
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