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 0.531738 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 0.138182 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 0.0379216 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 |