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D.8.2.3 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 from [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 from [4]..]/[1]

Note:
Applying elimpart to interred(i) may result in more substitutions. However, interred may be more expansive than elimpart for big ideals

Example:
 
LIB "presolve.lib";
ring s=0,(u,x,y,z),dp;
ideal i = xy2-xu4-x+y2,x2y2+z3+zy,y+z2+1,y+u2;
elimpart(i);
==> [1]:
==>    _[1]=u2-z2-1
==>    _[2]=u12-u2z+z3
==> [2]:
==>    _[1]=y
==>    _[2]=x
==> [3]:
==>    _[1]=u2+y
==>    _[2]=-u4+x
==> [4]:
==>    _[1]=u
==>    _[2]=0
==>    _[3]=0
==>    _[4]=z
==> [5]:
==>    _[1]=u
==>    _[2]=u4
==>    _[3]=-u2
==>    _[4]=z
i = interred(i); i;
==> i[1]=z2+y+1
==> i[2]=y2-x
==> i[3]=u2+y
==> i[4]=x3+z3+yz
elimpart(i);
==> [1]:
==>    _[1]=u2-z2-1
==>    _[2]=u12-u2z+z3
==> [2]:
==>    _[1]=x
==>    _[2]=y
==> [3]:
==>    _[1]=-y2+x
==>    _[2]=u2+y
==> [4]:
==>    _[1]=u
==>    _[2]=0
==>    _[3]=0
==>    _[4]=z
==> [5]:
==>    _[1]=u
==>    _[2]=u4
==>    _[3]=-u2
==>    _[4]=z
elimpart(i,2);
==> [1]:
==>    _[1]=z2+y+1
==>    _[2]=u2+y
==>    _[3]=y6+z3+yz
==> [2]:
==>    _[1]=x
==> [3]:
==>    _[1]=-y2+x
==> [4]:
==>    _[1]=u
==>    _[2]=0
==>    _[3]=y
==>    _[4]=z
==> [5]:
==>    _[1]=u
==>    _[2]=y2
==>    _[3]=y
==>    _[4]=z