The dual variety to a coincident root locus is the join of certain coincident root loci, as described in the paper by H. Lee and B. Sturmfels - Duality of multiple root loci - J. Algebra 446, 499-526, 2016.
i1 : X = coincidentRootLocus {5,3,2,2,1,1}
o1 = CRL(5,3,2,2,1,1)
o1 : CoincidentRootLocus
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i2 : dual X o2 = CRL(11,1,1,1) * CRL(13,1) * CRL(14) * CRL(14) (dual of CRL(5,3,2,2,1,1)) o2 : JoinOfCoincidentRootLoci |
In the example below, we apply some of the methods that are available for the objects returned by the method.
i3 : Y = dual coincidentRootLocus {4,2}
o3 = CRL(4,1,1) * CRL(6) (dual of CRL(4,2))
o3 : JoinOfCoincidentRootLoci
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i4 : ring Y
o4 = QQ[t , t , t , t , t , t , t ]
0 1 2 3 4 5 6
o4 : PolynomialRing
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i5 : coefficientRing Y o5 = QQ o5 : Ring |
i6 : dim Y o6 = 5 |
i7 : codim Y o7 = 1 |
i8 : degree Y o8 = 18 |
i9 : dual Y o9 = CRL(4,2) o9 : CoincidentRootLocus |
i10 : G = random Y
6 5 4 2 3 3 2 4 5 6
o10 = 3t + 12t t + 15t t + 80t t - 209655t t + 2582604t t - 8474625t
0 0 1 0 1 0 1 0 1 0 1 1
o10 : QQ[t , t ]
0 1
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i11 : member(G,Y) o11 = true |
i12 : ideal Y;
o12 : Ideal of QQ[t , t , t , t , t , t , t ]
0 1 2 3 4 5 6
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i13 : describe Y
o13 = Dual of the coincident root locus associated with the partition {4, 2} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({4, 1, 1},{6})
ambient: P^6 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6])
dim = 5
codim = 1
degree = 18
The defining polynomial has 3140 terms of degree 18
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