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

2013/07/09 by Ballico, Edoardo
#14N05 #14Q05 #15A69 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.2490

Abstract

Let σ_b(X_m,d(\mathbb C))(\mathbb R), b(m+1) < \binomm+dm, 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 \mathbb C. 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(\mathbb C))(\mathbb 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 sets 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.

Related

• On real typical ranks
• On the typical rank of real bivariate polynomials
• On the ranks and border ranks of symmetric tensors
• Gaps in the pairs (border rank,symmetric rank) for symmetric tensors
• The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables