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

## Synopsis

• Function: dual
• Usage:
dual X
• Inputs:
• Outputs:
• the dual variety to $X$

## Description

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 : Coincident root locus 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 : Join of 4 coincident root loci

In the example below, we apply some of the functions that are available for the returned objects.

 i3 : Y = dual coincidentRootLocus {4,2} o3 = CRL(4,1,1) * CRL(6) (dual of CRL(4,2)) o3 : Join of 2 coincident root loci i4 : ring Y o4 = QQ[t ..t ] 0 6 o4 : PolynomialRing 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 : Coincident root locus 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 i11 : member(G,Y) o11 = true i12 : ideal Y; o12 : Ideal of QQ[t ..t ] 0 6 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_6]) dim = 5 codim = 1 degree = 18 The defining polynomial has 3140 terms of degree 18