smallerMonomials -- returns the standard monomials smaller but of the same degree as given monomial(s)

• Usage:

  L = smallerMonomials M

  L = smallerMonomials(M, m)

• Inputs:
  • M, an ideal, $M$ should be a monomial ideal (an ideal generated by monomials)
  • m, a ring element, optional,
• Outputs:
  • L, a list, a list of lists: for each generator $m$ of $M$, the list of all monomials of the same degree as $m$, not in the monomial ideal and smaller than that generator in the term order of the ambient ring. If instead $m$ is given, the list of the standard monomials of the same degree, smaller than $m$, is returned.

i1 : R = ZZ/32003[a..d];

i2 : M = ideal (a^2, b^2, a*b*c);

o2 : Ideal of R

i3 : L1 = smallerMonomials M

                       2                  2               2                
o3 = {{a*b, a*c, b*c, c , a*d, b*d, c*d, d }, {a*c, b*c, c , a*d, b*d, c*d,
     ------------------------------------------------------------------------
      2       2     2   3                        2      2     2     2   3
     d }, {a*c , b*c , c , a*b*d, a*c*d, b*c*d, c d, a*d , b*d , c*d , d }}

o3 : List

i4 : smallerMonomials(M, b^2)

                 2                  2
o4 = {a*c, b*c, c , a*d, b*d, c*d, d }

o4 : List