5.1.55 highcorner

Syntax:

highcorner ( ideal_expression )
highcorner ( module_expression )

Type:

poly, resp. vector

Purpose:

returns the smallest monomial not contained in the ideal, resp. module, generated by the initial terms of the given generators. If the generators are a standard basis, this is also the smallest monomial not contained in the ideal, resp. module.
If the ideal, resp. module, is not zero-dimensional, 0 is returned.
The command works also in global orderings, but is not very useful there.

Note:

Let the ideal I be given by a standard basis. Then highcorner(I) returns 0 if and only if dim(I)>0 or dim(I)=-1. Otherwise it returns the smallest monomial m not in I which has the following properties (with the variables of the basering):

• if then does not divide m (hence, m=1 if the ordering is global)
• given any set of generators of I, let be obtained from by deleting the terms divisible by for all i with . Then generate I.

Example:

ring r=0,(x,y),ds; ideal i=x3,x2y,y3; highcorner(std(i)); ==> xy2 highcorner(std(ideal(1))); ==> 0