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# realRankBoundary -- algebraic boundaries among typical ranks for real binary forms

## Synopsis

• Usage:
realRankBoundary(n,i)
• Inputs:
• Optional inputs:
• Variable => ..., default value "t", specify a name for a variable
• Outputs:
• the irreducible components of the real algebraic boundary that separates real rank $i$ from the other typical ranks in the real projective space $\mathbb{P}^n=\mathbb{P}(Sym^n(\mathbb{R}^2))$ of binary forms of degree $n$

## Description

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 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

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 ------------------------------------------------------------------------ CRL(4,2))} o3 : List 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 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 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 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

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) ------------------------------------------------------------------------ (dual of CRL(4,3)), CRL(4,1,1,1) * CRL(7) (dual of CRL(5,2))} o8 : List 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 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 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 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 ----------------------------------------------------------------------- of CRL(5,2)), CRL(2,1,1,1,1,1)} o12 : List 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 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 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