# ciResDeg -- compute a regularity index and partial degrees of the residual resultant over a complete intersection

## Synopsis

• Usage:
ciResDeg(List, List)
• Inputs:
• d, a list, a list of element in the ambient ring corresponding to the degrees of a system of polynomials $d_0,...,d_n$
• k, a list, a list of element in the ambient ring corresponding to the degrees of a complete intersection $k_1,...,k_m$, where m=<n
• Outputs:
• List, a list, a list of element consisting of a regularity index and partial degree of homogeneity

## Description

Given a system of polynomials $f_0,...,f_n$ of degree $d_0,...,d_n$ that are contained in a complete intersection $g_1,...,g_m$ of degree $k_1,...,k_m$, this function returns the regularity index used to form the matrix associated to the residual resultant over a complete intersection and then all the partial degrees of this resultant with respect to the coefficients of $f_0,f_1,..,f_n$.

 i1 : R=ZZ[d_0..d_3,k_1,k_2] o1 = R o1 : PolynomialRing i2 : L=ciResDeg({d_0,d_1,d_2,d_3},{k_1,k_2}) 2 o2 = {d + d + d + d - 3k - 3, {d d d - d k k - d k k - d k k + k k 0 1 2 3 2 1 2 3 1 1 2 2 1 2 3 1 2 1 2 ------------------------------------------------------------------------ 2 2 2 + k k , d d d - d k k - d k k - d k k + k k + k k , d d d - d k k 1 2 0 2 3 0 1 2 2 1 2 3 1 2 1 2 1 2 0 1 3 0 1 2 ------------------------------------------------------------------------ 2 2 - d k k - d k k + k k + k k , d d d - d k k - d k k - d k k + 1 1 2 3 1 2 1 2 1 2 0 1 2 0 1 2 1 1 2 2 1 2 ------------------------------------------------------------------------ 2 2 k k + k k }} 1 2 1 2 o2 : List