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Fix integers m≥5 and d≥3. Let f be a degree d homogeneous polynomial in m+1 variables. Here, we prove that f is the sum of at most d·⌈(m+dm)/(m+1)⌉d-powers of linear forms (of course, this inequality is nontrivial only if m≫d.

An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables