Define Rn,i as the interior of the set {F∈Symn(ℝ2) : realrank(F) = i}. Then Rn,i is a semi-algebraic set which is non-empty exactly when (n+1)/2 ≤i≤n (in this case we say that i is a typical rank); see the paper by G. Blekherman - Typical real ranks of binary forms - Found. Comput. Math. 15, 793-798, 2015. The topological boundary ∂(Rn,i) is the set-theoretic difference of the closure of Rn,i minus the interior of the closure of Rn,i. In the range (n+1)/2 ≤i≤n-1, it is a semi-algebraic set of pure codimension one. The (real) algebraic boundary ∂alg(Rn,i) is defined as the Zariski closure of the topological boundary ∂(Rn,i). This is viewed as a hypersurface in ℙ(Symn(ℝ2)) and the method returns its irreducible components over C.
In the case i = n, the algebraic boundary ∂alg(Rn,n) is the discriminant hypersurface; see the paper by A. Causa and R. Re - On the maximum rank of a real binary form - Ann. Mat. Pura Appl. 190, 55-59, 2011; see also the paper by P. Comon, G. Ottaviani - On the typical rank of real binary forms - Linear Multilinear Algebra 60, 657-667, 2012.
i1 : time D77 = realRankBoundary(7,7)
-- used 0.50038 seconds
o1 = CRL(2,1,1,1,1,1)
o1 : CoincidentRootLocus
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i2 : describe D77
o2 = Coincident root locus associated with the partition {2, 1, 1, 1, 1, 1} defined over QQ
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 12
The singular locus is the union of the coincident root loci associated with the partitions:
({2, 2, 1, 1, 1},{3, 1, 1, 1, 1})
The defining polynomial has 1103 terms of degree 12
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In the opposite extreme case, i = ceiling((n+1)/2), the algebraic boundary ∂alg(Rn,i) has been described in the paper by H. Lee and B. Sturmfels - Duality of multiple root loci - J. Algebra 446, 499-526, 2016. It is irreducible if n is odd, and has two irreducible components if n is even.
i3 : time D64 = realRankBoundary(6,4)
-- used 2.76743 seconds
o3 = {CRL(5,1) * CRL(5,1) (dual of CRL(3,3)), CRL(4,1,1) * CRL(6) (dual of
------------------------------------------------------------------------
CRL(4,2))}
o3 : List
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i4 : describe first D64
o4 = Dual of the coincident root locus associated with the partition {3, 3} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({5, 1},{5, 1})
ambient: P^6 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6])
dim = 5
codim = 1
degree = 12
The defining polynomial has 560 terms of degree 12
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i5 : describe last D64
o5 = 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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i6 : time D74 = realRankBoundary(7,4)
-- used 0.444119 seconds
o6 = CRL(6,1) * CRL(7) * CRL(7) (dual of CRL(3,2,2))
o6 : JoinOfCoincidentRootLoci
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i7 : describe D74
o7 = Dual of the coincident root locus associated with the partition {3, 2, 2} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions:
({6, 1},{7},{7})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 24
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In the next example, we compute the irreducible components of the algebraic boundaries ∂alg(R7,5) and ∂alg(R7,6).
i8 : time D75 = realRankBoundary(7,5)
-- used 0.229553 seconds
o8 = {CRL(6,1) * CRL(7) * CRL(7) (dual of CRL(3,2,2)), CRL(5,1,1) * CRL(6,1)
------------------------------------------------------------------------
(dual of CRL(4,3)), CRL(4,1,1,1) * CRL(7) (dual of CRL(5,2))}
o8 : List
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i9 : describe D75_0
o9 = Dual of the coincident root locus associated with the partition {3, 2, 2} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions:
({6, 1},{7},{7})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 24
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i10 : describe D75_1
o10 = Dual of the coincident root locus associated with the partition {4, 3} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({5, 1, 1},{6, 1})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 36
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i11 : describe D75_2
o11 = Dual of the coincident root locus associated with the partition {5, 2} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({4, 1, 1, 1},{7})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 24
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i12 : time D76 = realRankBoundary(7,6)
-- used 0.000298063 seconds
o12 = {CRL(5,1,1) * CRL(6,1) (dual of CRL(4,3)), CRL(4,1,1,1) * CRL(7) (dual
-----------------------------------------------------------------------
of CRL(5,2)), CRL(2,1,1,1,1,1)}
o12 : List
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i13 : describe D76_0
o13 = Dual of the coincident root locus associated with the partition {4, 3} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({5, 1, 1},{6, 1})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 36
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i14 : describe D76_1
o14 = Dual of the coincident root locus associated with the partition {5, 2} defined over QQ
which coincides with the join of the coincident root loci associated with the partitions: ({4, 1, 1, 1},{7})
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 24
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i15 : describe D76_2
o15 = Coincident root locus associated with the partition {2, 1, 1, 1, 1, 1} defined over QQ
ambient: P^7 = Proj(QQ[t_0, t_1, t_2, t_3, t_4, t_5, t_6, t_7])
dim = 6
codim = 1
degree = 12
The singular locus is the union of the coincident root loci associated with the partitions:
({2, 2, 1, 1, 1},{3, 1, 1, 1, 1})
The defining polynomial has 1103 terms of degree 12
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