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5.1.56 indepSet
Syntax:
indepSet ( ideal_expression )
Type:
- intvec
Purpose:
- computes a maximal set U of independent variables (in the sense defined in
the note below) of the ideal given by
a standard basis. If
v is the result then v[i] is 1 if and
only if the i-th variable of the ring, x(i), is an independent
variable. Hence, the set U consisting of all variables x(i) with
v[i]=1 is a maximal independent set.
Note:
- U is a set of independent variables for I if and only if
![$I \cap K[U]=(0)$](sing_136.png) ,i.e., eliminating the remaining variables gives (0).
U is maximal if dim(I)=#U.
Syntax:
indepSet ( ideal_expression, int_expression )
Type:
- list
Purpose:
- computes a list of all maximal independent sets of the leading ideal
(if the flag is 0), resp. of all those sets of independent variables
of the leading ideal which cannot be enlarged.
Example:
| | ring r=32003,(x,y,u,v,w),dp;
ideal I=xyw,yvw,uyw,xv;
attrib(I,"isSB",1);
indepSet(I);
==> 1,1,1,0,0
eliminate(I,vw);
==> _[1]=0
indepSet(I,0);
==> [1]:
==> 1,1,1,0,0
==> [2]:
==> 0,1,1,1,0
==> [3]:
==> 1,0,1,0,1
==> [4]:
==> 0,0,1,1,1
indepSet(I,1);
==> [1]:
==> 1,1,1,0,0
==> [2]:
==> 0,1,1,1,0
==> [3]:
==> 1,0,1,0,1
==> [4]:
==> 0,0,1,1,1
==> [5]:
==> 0,1,0,0,1
eliminate(I,xuv);
==> _[1]=0
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See
ideal;
std.
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