          ##### 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=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=x*Dx^2-y*Dy^2+(c)*Dx ==> pGBid=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=(2*x*y)*Dx*Dy+(x*y+y^2-y)*Dy^2+(-c*x-2*x)*Dx+(-2*y)*Dy+2 ==> rGBid=(x)*Dx^2+(-y)*Dy^2+(c)*Dx ==> rGBid=(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 ```          User manual for Singular version 4.3.2, 2023, generated by texi2html.