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

- 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

## See also

- eliminationMatrix -- returns a matrix that represents the image of the map
- ciResDegGH -- compute a regularity index used for the residual resultant over a complete intersection

## For the programmer

The object ciResDeg is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/EliminationMatrices.m2:1494:0.