Singular

5.1.81 liftstd

Syntax:

liftstd ( ideal_expression, matrix_name[, module_name][, string_expression ][, ideal_expression ])
liftstd ( module_expression, matrix_name[, module_name][, string_expression ][, module_expression ])

Type:

ideal or module

Purpose:

returns a standard basis of an ideal or module and the transformation matrix from the given ideal, resp. module, to the standard basis.
That is, if m is the ideal or module, sm the standard basis returned by liftstd, and T the transformation matrix (sm=liftstd(m,T)) then matrix(sm)=matrix(m)*T and sm=ideal(matrix(m)*T), resp. sm=module(matrix(m)*T). If working in a quotient ring, then matrix(sm)=reduce(matrix(m)*T,0) and sm=reduce(ideal(matrix(m)*T),0).
If a module name is given as a third argument, the syzygy module will be returned. (sm=liftstd(m,T,s) then additional matrix(m)*matrix(s)=0).
An optional string argument specifies the Groebner base algorithm to use. Possible values are "std" and "slimgb".
Given an optional last argument (say n), the algorithm computes a standard bases of (m+n), syzygies of m modulo n, and the transformation matrix only for m. These are relative transformation matrix resp. the syzygy module of n modulo m. (For syzygies, the same can be achieved using modulo.)

Example:

ring R=0,(x,y,z),dp; poly f=x3+y7+z2+xyz; ideal i=jacob(f); matrix T; ideal sm=liftstd(i,T); sm; ==> sm[1]=xy+2z ==> sm[2]=3x2+yz ==> sm[3]=yz2+3048192z3 ==> sm[4]=3024xz2-yz2 ==> sm[5]=y2z-6xz ==> sm[6]=3097158156288z4+2016z3 ==> sm[7]=7y6+xz print(T); ==> 0,1,T[1,3], T[1,4],y, T[1,6],0, ==> 0,0,-3x+3024z,3x, 0, T[2,6],1, ==> 1,0,T[3,3], T[3,4],-3x,T[3,6],0 matrix(sm)-matrix(i)*T; ==> _[1,1]=0 ==> _[1,2]=0 ==> _[1,3]=0 ==> _[1,4]=0 ==> _[1,5]=0 ==> _[1,6]=0 ==> _[1,7]=0 module s; sm=liftstd(i,T,s); print(s); ==> -xy-2z,0, s[1,3],s[1,4],xyz+2z2, -14y5z+x2z, ==> 0, -xy-2z,s[2,3],s[2,4],-3x2y-6xz,-3x3+2z2, ==> 3x2+yz,7y6+xz,7y6+xz,s[3,4],21xy6-yz2,21x2y5-xz2 matrix(i)*matrix(s); ==> _[1,1]=0 ==> _[1,2]=0 ==> _[1,3]=0 ==> _[1,4]=0 ==> _[1,5]=0 ==> _[1,6]=0