D.7.1.3 elimpart

Procedure from library presolve.lib (see presolve_lib).

Usage:

elimpart(i [,n,e] ); i=ideal, n,e=integers
n : only the first n vars are considered for substitution,
e =0: substitute from linear part of i (same as elimlinearpart)
e!=0: eliminate also by direct substitution
(default: n = nvars(basering), e = 1)

Return:

list of 5 objects:

[1]: ideal obtained by substituting from the first n variables those from i, which appear in the linear part of i (or, if e!=0, which can be expressed directly in the remaining vars) [2]: ideal, variables which have been substituted [3]: ideal, i-th element defines substitution of i-th var in [2] [4]: ideal of variables of basering, substituted ones are set to 0 [5]: ideal, describing the map from the basering, say k[x(1..m)], to itself onto k[..variables fom [4]..] and [1] is the image of i

The ideal i is generated by [1] and [3] in k[x(1..m)], the map [5] maps [3] to 0, hence induces an isomorphism

k[x(1..m)]/i -> k[..variables fom [4]..]/[1]

Note:

If the basering has ordering (c,dp), this is faster for big ideals, since it avoids internal ring change and mapping.

Example:

LIB "presolve.lib";
ring s=0,(x,y,z),dp;
ideal i =x2+y2,x2+y+1;
elimpart(i,3,0);
→ [1]:
→    _[1]=y2-y-1
→    _[2]=x2+y+1
→ [2]:
→    _[1]=0
→ [3]:
→    _[1]=0
→ [4]:
→    _[1]=x
→    _[2]=y
→    _[3]=z
→ [5]:
→    _[1]=x
→    _[2]=y
→    _[3]=z
elimpart(i,3,1);
→ [1]:
→    _[1]=x4+3x2+1
→ [2]:
→    _[1]=y
→ [3]:
→    _[1]=x2+y+1
→ [4]:
→    _[1]=x
→    _[2]=0
→    _[3]=z
→ [5]:
→    _[1]=x
→    _[2]=-x2-1
→    _[3]=z