# numgensByCodim -- maximum number of generators of localizations of a monomial ideal

## Synopsis

• Usage:
d = numgensByCodim(I,k)
L = numgensByCodim(I)
• Inputs:
• I, ,
• k, an integer, an integer between 1 and the dimension of the ring
• Outputs:
• d, an integer, the maximum number of generators of I localized at a prime P of codimension k.
• L, a list, a list of the numbers of generators for each codimension from 1 to the dimension of the ring

## Description

Because I is monomial, we can check the number of generators of I localized at a prime P over only monomial primes P.

 i1 : R = QQ[x_0..x_4]; i2 : I = monomialIdeal{x_0^2,x_1*x_2,x_3*x_4^2} 2 2 o2 = monomialIdeal (x , x x , x x ) 0 1 2 3 4 o2 : MonomialIdeal of R i3 : numgensByCodim(I,2) o3 = 1 i4 : numgensByCodim I o4 = {1, 1, 3, 3, 3} o4 : List