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# CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci

- 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

## Ways to use this method:

- CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CoincidentRootLoci/documentation.m2:370:0.