# standardMonomials -- computes standard monomials

## Synopsis

• Usage:
L = standardMonomials M
L = standardMonomials(d, M)
• Inputs:
• M, an ideal, M should be a monomial ideal
• d, a list, a degree
• Outputs:
• L, a list, L is a list of lists of standard monomials for the ideal $M$, one for each generator of $M$

## Description

A monomial $m$ is standard with respect to a monomial ideal $M$ and a generator $g$ of $M$ if $m$ is of the same degree as $g$ but is not an element of $M$.

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

 i1 : R = ZZ/32003[a..d]; i2 : M = ideal (a^2, a*b, b^3, a*c); o2 : Ideal of R i3 : L1 = standardMonomials M 2 2 2 2 2 2 o3 = {{b , b*c, c , a*d, b*d, c*d, d }, {b , b*c, c , a*d, b*d, c*d, d }, ------------------------------------------------------------------------ 2 2 3 2 2 2 2 2 3 2 2 {b c, b*c , c , b d, b*c*d, c d, a*d , b*d , c*d , d }, {b , b*c, c , ------------------------------------------------------------------------ 2 a*d, b*d, c*d, d }} o3 : List i4 : standardMonomials({3}, M) 2 2 3 2 2 2 2 2 3 o4 = {b c, b*c , c , b d, b*c*d, c d, a*d , b*d , c*d , d } o4 : List

Inputting an integer $d$ (or degree $d$) and an ideal gives the standard monomials for the specified ideal in degree $d$.

 i5 : standardMonomials(2, M) 2 2 2 o5 = {b , b*c, c , a*d, b*d, c*d, d } o5 : List