D.7.1.4 elimpartanyr

Procedure from library presolve.lib (see presolve_lib).

Usage:

elimpartanyr(i [,p,e] ); i=ideal, p=polynomial, e=integer
p: product of vars to be eliminated,
e =0: substitute from linear part of i (same as elimlinearpart)
e!=0: eliminate also by direct substitution
(default: p=product of all vars, e=1)

Return:

list of 6 objects:

[1]: (interreduced) ideal obtained by substituting from i those vars appearing in p, which occur in the linear part of i (or which can be expressed directly in the remaining variables, if e!=0) [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 [6]: int, # of vars considered for substitution (= # of factors of p)

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:

the proc uses execute to create a ring with ordering dp and vars placed correctly and then applies elimpart.

Example:

LIB "presolve.lib";
ring s=0,(x,y,z),dp;
ideal i = x3+y2+z,x2y2+z3,y+z+1;
elimpartanyr(i,z);
→ [1]:
→    _[1]=x3+y2-y-1
→    _[2]=x2y2-y3-3y2-3y-1
→ [2]:
→    _[1]=z
→ [3]:
→    _[1]=y+z+1
→ [4]:
→    _[1]=0
→    _[2]=x
→    _[3]=y
→ [5]:
→    _[1]=-y-1
→    _[2]=x
→    _[3]=y
→ [6]:
→    1