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

## Synopsis

• Operator: *
• Usage:
X * Y
projectiveJoin(X,Y)
• Inputs:
• 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