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

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