On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank

Let σ b (X m, d ()) () σb(Xm,d(C))(R), b (m + 1) < m + d m b(m+1) < {m+d m}, denote the set of all degree d real homogeneous polynomials in m + 1 variables (i.e., real symmetric tensors of format (m + 1) × ⋯ × (m + 1), d times) which have border rank b over . It has a partition into manifolds of real dimension ≤ b(m + 1)-1 in which the real rank is constant. A typical rank of σ b (X m, d ()) () σb(Xm,d(C))(R) is a rank associated to an open part of dimension b(m + 1) - 1. Here, we classify all typical ranks when b ≤ 7 and d, m are not too small. For a larger set of (m, d, b), we prove that b and b + d - 2 are the two first typical ranks. In the case m = 1 (real bivariate polynomials), we prove that d (the maximal possible a priori value of the real rank) is a typical rank for every b. © 2014 Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.