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5.1.53 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 $x_i$the variables of the basering):
  • if $x_i>1$ then $x_i$does not divide m (hence, m=1 if the ordering is global)
  • given any set of generators $f_1,\dots,f_k$ of I, let $f'_i$ be obtained from $f_i$ by deleting the terms divisible by $x_i\cdot m$ for all i with $x_i<1$. Then $f'_1,\dots,f'_k$ generate I.
Example:
 
ring r=0,(x,y),ds;
ideal i=x3,x2y,y3;
highcorner(std(i));
==> xy2
highcorner(std(ideal(1)));
==> 0
See dim; std; vdim.