CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci

Synopsis

• Operator: *

• Usage:

  X * Y

  projectiveJoin(X,Y)

• Inputs:
  • X, a coincident root locus
  • Y, a coincident root locus
• Outputs:
  • the projective join of $X$ and $Y$

Description

A partition of a number $n$ is a hook if at most one part is not 1. The inputs of this method are required to be coincident root loci associated with hook partitions of $n$. In this case, the returned object is the dual of a certain coincident root locus; see the paper by H. Lee and B. Sturmfels - Duality of multiple root loci - J. Algebra 446, 499-526, 2016.

i1 : X = coincidentRootLocus {11,1,1,1,1}

o1 = CRL(11,1,1,1,1)

o1 : Coincident root locus

i2 : Y = coincidentRootLocus {13,1,1}

o2 = CRL(13,1,1)

o2 : Coincident root locus

i3 : X * Y

o3 = CRL(11,1,1,1,1) * CRL(13,1,1) (dual of CRL(6,4,1,1,1,1,1))

o3 : Join of 2 coincident root loci

i4 : X * Y * Y

o4 = CRL(11,1,1,1,1) * CRL(13,1,1) * CRL(13,1,1) (dual of CRL(6,4,4,1))

o4 : Join of 3 coincident root loci

More generally, if $I_1,I_2,\ldots$ is a sequence of homogeneous ideals (resp. parameterizations) of projective varieties $X_1,X_2,\ldots \subset \mathbb{P}^n$, then projectiveJoin(I_1,I_2,...) is the ideal of the projective join $X_1\,*\,X_2\,*\,\cdots \subset \mathbb{P}^n$.

i5 : I = ideal coincidentRootLocus {4}

             2                                    2                       2
o5 = ideal (t  - t t , t t  - t t , t t  - t t , t  - t t , t t  - t t , t  -
             3    2 4   2 3    1 4   1 3    0 4   2    0 4   1 2    0 3   1  
     ------------------------------------------------------------------------
     t t )
      0 2

o5 : Ideal of QQ[t ..t ]
                  0   4

i6 : projectiveJoin(I,I)

            3                2    2
o6 = ideal(t  - 2t t t  + t t  + t t  - t t t )
            2     1 2 3    0 3    1 4    0 2 4

o6 : Ideal of QQ[t ..t ]
                  0   4

See also

• dual(CoincidentRootLocus) -- the projectively dual to a coincident root locus