# regularityVar -- computes the Castelnuovo-Mumford regularity of homogeneous ideals in terms of Betti numbers, with respect to some of the variables of the ring

## Synopsis

• Usage:
regularityVar(l,I)
• Inputs:
• l, a list, list of variables of the polynomial ring R to take into account for computing the Castelnuovo-Mumford regularity
• I, an ideal, ideal of a polynomial ring

## Description

regularityVar computes the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring by computing the shifts and degrees of generators in a minimal free resolution of the homogeneous ideal.

The list of variables l contains the variables of the ring having degree 1. Those variables on the ring not in l have automatically degree 0, as well as the the elements on the coefficient ring

 i1 : R=QQ[a..i,x,y,z] o1 = R o1 : PolynomialRing i2 : f1 = a*x+b*y+c*z o2 = a*x + b*y + c*z o2 : R i3 : f2 = d*x+e*y+f*z o3 = d*x + e*y + f*z o3 : R i4 : f3 = g*x+h*y+i*z o4 = g*x + h*y + i*z o4 : R i5 : I = ideal(f1,f2,f3) o5 = ideal (a*x + b*y + c*z, d*x + e*y + f*z, g*x + h*y + i*z) o5 : Ideal of R i6 : l = {x,y,z} o6 = {x, y, z} o6 : List i7 : regularityVar (l,I) o7 = 1