# ciResDegGH -- compute a regularity index used for the residual resultant over a complete intersection

## Synopsis

• Usage:
ciResDegGH(r, m, v)
• Inputs:
• v, a list, list 'var' of variables with respect to which the polynomials are homogeneous and from which one wants to remove these variables
• r, , a single row matrix describing the base locus
• m, , a matrix corresponding to the decomposition of a polynomial system over the base locus
• Outputs:
• an integer, a regularity index to form the residual resultant-

## Description

This function is similar to the first element in the list returned by the function ciResDeg but with arguments that are identical to the ones used with the function eliminationMatrix using the Strategy CM2Residual.

 i1 : R=QQ[a_0,a_1,a_2,a_3,a_4,b_0,b_1,b_2,b_3,b_4,c_0,c_1,c_2,c_3,c_4,x,y,z] o1 = R o1 : PolynomialRing i2 : G=matrix{{z,x^2+y^2}} o2 = | z x2+y2 | 1 2 o2 : Matrix R <--- R i3 : H=matrix{{a_0*z+a_1*x+a_2*y,b_0*z+b_1*x+b_2*y,c_0*z+c_1*x+c_2*y},{a_3,b_3,c_3}} o3 = | a_1x+a_2y+a_0z b_1x+b_2y+b_0z c_1x+c_2y+c_0z | | a_3 b_3 c_3 | 2 3 o3 : Matrix R <--- R i4 : ciResDegGH({x,y,z},G,H) o4 = 2

## See also

• eliminationMatrix -- returns a matrix that represents the image of the map
• ciResDeg -- compute a regularity index and partial degrees of the residual resultant over a complete intersection

## For the programmer

The object ciResDegGH is .