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

## Synopsis

• Usage:
L = smallerMonomials M
L = smallerMonomials(M, m)
• Inputs:
• M, an ideal, $M$ should be a monomial ideal (an ideal generated by monomials)
• m, , 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