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Acta Mathematica Vietnamica

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

icon-email Edoardo Ballico

Abstract

Let σb(Xm,d(C))(R), b(m+1)<(m+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 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 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 b7 and d,m are not too small. For a larger set of (m,d,b) we prove that b and b+d2 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.