next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
CoincidentRootLoci :: realRankBoundary

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

Synopsis

Description

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 0.611583 seconds

o1 = CRL(2,1,1,1,1,1)

o1 : CoincidentRootLocus
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

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 3.24222 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
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
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
i6 : time D74 = realRankBoundary(7,4)
     -- used 0.516661 seconds

o6 = CRL(6,1) * CRL(7) * CRL(7) (dual of CRL(3,2,2))

o6 : JoinOfCoincidentRootLoci
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

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 0.297916 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
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
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
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
i12 : time D76 = realRankBoundary(7,6)
     -- used 0.000313117 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
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
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
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

See also

Ways to use realRankBoundary :