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# 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.

## Description

Inputting an ideal $M$ returns the smaller monomials of each of the given generators of the ideal.

    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

## See also

- tailMonomials -- find tail monomials
- standardMonomials -- computes standard monomials

## Ways to use smallerMonomials:

- `smallerMonomials(Ideal)`
- `smallerMonomials(Ideal,RingElement)`

## For the programmer

The object smallerMonomials is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GroebnerStrata.m2:524:0.