# tailMonomials -- find tail monomials

## Synopsis

• Usage:
L = tailMonomials M
L = tailMonomials(M, m)
• Inputs:
• M, an ideal, $M$ should be a monomial ideal (an ideal generated by monomials)
• m, , optional, only return a single list of the tail monomials for this monomial
• Optional inputs:
• AllStandard => , default value false, which monomials should be considered tail monomials of a monomial $m$: either all standard monomials of a given degree, or all monomials smaller than $m$ in the given term order (but still of the same degree)
• Outputs:
• L, a list, a list of lists: for each generator $m$ of $M$, the list of all tail monomials If instead $m$ is given, the list of the tail monomials of $m$ is returned

## Description

Inputting an ideal $M$ generated by monomials returns a list of lists of tail monomialsfor each generator of $M$ (in the smae order).

 i1 : R = ZZ/32003[a..d]; i2 : M = ideal (a^2, b^2, a*b*c); o2 : Ideal of R i3 : tailMonomials 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 : tailMonomials(M, AllStandard => true) 2 2 2 o4 = {{a*b, a*c, b*c, c , a*d, b*d, c*d, d }, {a*b, a*c, b*c, c , a*d, b*d, ------------------------------------------------------------------------ 2 2 2 3 2 2 2 2 c*d, d }, {a*c , b*c , c , a*b*d, a*c*d, b*c*d, c d, a*d , b*d , c*d , ------------------------------------------------------------------------ 3 d }} o4 : List i5 : tailMonomials(M, b^2) 2 2 o5 = {a*c, b*c, c , a*d, b*d, c*d, d } o5 : List i6 : tailMonomials(M, b^2, AllStandard=>true) 2 2 o6 = {a*b, a*c, b*c, c , a*d, b*d, c*d, d } o6 : List