ON THE UNIQUENESS FOR THE SYMMETRIC TENSOR RANK OF TRIVARIATE POLYNOMIALS; A LOCAL UNIQUENESS FOR MULTIVARIATE POLYNOMIALS

Abstract

For all integers m ≥ 2 and d ≥ 3 and x > 0. Let νd : P m → P N , N := m+d m � − 1, be the Veronese embedding. We discuss the uniqueness (only for trivariate polynomials) and the local uniqueness of a decomposition of a polynomial into powers of linear forms in the following sense. Take P ∈ P N. Let S(m,x,d,P) be the set of all S ⊂ P m such that ♯(S) = x, P ∈ hνd(S)i (where h iis the linear span), and P /