Singular

5.1.26 eliminate

Syntax:

eliminate ( ideal_expression, product_of_ring_variables )
eliminate ( module_expression, product_of_ring_variables )
eliminate ( ideal_expression, intvec_expression )
eliminate ( module_expression, intvec_expression )
eliminate ( ideal_expression, product_of_ring_variables, intvec_hilb )
eliminate ( module_expression, product_of_ring_variables, intvec_hilb )

Type:

the same as the type of the first argument

Purpose:

eliminates variables occurring as factors/entries of the second argument from an ideal (resp. a submodule of a free module), by intersecting it (resp. each component of the submodule) with the subring not containing these variables.
eliminate does not need a special ordering nor a standard basis as input.

Note:

Since elimination is expensive, for homogeneous input it might be useful first to compute the Hilbert function of the ideal (first argument) with a fast ordering (e.g., dp). Then make use of it to speed up the computation: a Hilbert-driven elimination uses the intvec provided as the third argument.
If the ideal (resp. module) is not homogeneous with weights 1, this intvec will be silently ignored.

Example:

ring r=32003,(x,y,z),dp; ideal i=x2,xy,y5; eliminate(i,x); ==> _[1]=y5 ring R=0,(x,y,t,s,z),dp; ideal i=x-t,y-t2,z-t3,s-x+y3; eliminate(i,ts); ==> _[1]=y2-xz ==> _[2]=xy-z ==> _[3]=x2-y ideal j=x2,xy,y2; intvec v=hilb(std(j),1); eliminate(j,y,v); ==> _[1]=x2