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7.7.20.0. ratstd
Procedure from libraryratgb.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 the elimination block ordering for the variables
in the second block
- Note:
- the output ring C is commutative. The ideal
rGBidin C represents the rational form of the output idealpGBidin the basering. - During the computation, the D-dimension of I and the corresponding dimension as K(x)-vector space of I are computed and printed out. - Setting optional integer eng to 1, slimgb is taken as Groebner engine - Display:
- In order to see the steps of the computation, set printlevel to >=2
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 ==> the D-dimension is 0 ==> the K-dimension is 4 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) //--- Now, let us compute a K(x,y) basis explicitly print(matrix(kbase(rGBid))); ==> // ** rGBid is no standard basis ==> Dy^2,Dy,Dx,1 |