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

 

Abstract

Let σ b ( X m , d ( C ) ) ( R ) , 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 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 ( 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.

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