Define ${\mathcal{R}}_{n,i}$ as the interior of the set $\{F\in Sym^n(\mathbb{R}^2) : realrank(F) = i\}$. Then ${\mathcal{R}}_{n,i}$ is a semi-algebraic set which is non-empty exactly when $(n+1)/2 \leq i\leq n$ (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 $\partial({\mathcal{R}}_{n,i})$ is the set-theoretic difference of the closure of ${\mathcal{R}}_{n,i}$ minus the interior of the closure of ${\mathcal{R}}_{n,i}$. In the range $(n+1)/2 \leq i\leq n-1$, it is a semi-algebraic set of pure codimension one. The (real) algebraic boundary $\partial_{alg}({\mathcal{R}}_{n,i})$ is defined as the Zariski closure of the topological boundary $\partial({\mathcal{R}}_{n,i})$. This is viewed as a hypersurface in $\mathbb{P}(Sym^n(\mathbb{R}^2))$ and the method returns its irreducible components over $\mathbb{C}$.
In the case $i = n$, the algebraic boundary $\partial_{alg}({\mathcal{R}}_{n,n})$ 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 : D77 = realRankBoundary(7,7)
o1 = CRL(2,1,1,1,1,1)
o1 : Coincident root locus
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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_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 $\partial_{alg}({\mathcal{R}}_{n,i})$ 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 : D64 = realRankBoundary(6,4)
o3 = {CRL(5,1) * CRL(5,1) (dual of CRL(3,3)), CRL(4,1,1) * CRL(6) (dual of
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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_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_6])
dim = 5
codim = 1
degree = 18
The defining polynomial has 3140 terms of degree 18
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i6 : D74 = realRankBoundary(7,4)
o6 = CRL(6,1) * CRL(7) * CRL(7) (dual of CRL(3,2,2))
o6 : Join of 3 coincident root loci
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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_7])
dim = 6
codim = 1
degree = 24
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In the next example, we compute the irreducible components of the algebraic boundaries $\partial_{alg}({\mathcal{R}}_{7,5})$ and $\partial_{alg}({\mathcal{R}}_{7,6})$.
i8 : D75 = realRankBoundary(7,5)
o8 = {CRL(6,1) * CRL(7) * CRL(7) (dual of CRL(3,2,2)), CRL(5,1,1) * CRL(6,1)
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(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_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_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_7])
dim = 6
codim = 1
degree = 24
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i12 : D76 = realRankBoundary(7,6)
o12 = {CRL(5,1,1) * CRL(6,1) (dual of CRL(4,3)), CRL(4,1,1,1) * CRL(7) (dual
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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_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_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_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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