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CoincidentRootLoci :: CoincidentRootLocus * CoincidentRootLocus

CoincidentRootLocus * CoincidentRootLocus -- projective join of coincident root loci

Synopsis

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

o2 = CRL(13,1,1)

o2 : CoincidentRootLocus
i3 : time X * Y
     -- used 0.575931 seconds

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

o3 : JoinOfCoincidentRootLoci
i4 : time X * Y * Y
     -- used 0.143298 seconds

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

o4 : JoinOfCoincidentRootLoci

More generally, if I1,I2,... is a sequence of homogeneous ideals (resp. parameterizations) of projective varieties X1,X2,...⊂ℙn, then projectiveJoin(I_1,I_2,...) is the ideal of the projective join X1 * X2 * …⊂ℙn.

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 , t , t , t ]
                  0   1   2   3   4
i6 : time projectiveJoin(I,I)
     -- used 0.0414323 seconds

            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 , t , t , t ]
                  0   1   2   3   4

See also