On the order of approximation in approximative triadic decompositions of tensors (Q1123957)

Based on \textit{A. Alder's} result [Grenzrang und Grenzkomplexität aus algebraischer und topologischer Sicht, Dissertation, Univ. Zürich (1984)] on the equivalence of algebraic and topological border rank of tensors t over algebraically closed ground fields k, the author gives an upper bound for the order of approximation in approximative triadic decomposition \(t=\sum^{r}_{\rho =1}u_{\rho}\otimes V_{\rho}\otimes w_{\rho}\) where \(u_{\rho}\in k^ n\), \(v_{\rho}\in k^ n\), \(w_{\rho}\in k^ n\). If \(R_ h(t)\) is defined to be the approximative rank of order of some t, Ṟ(t) is the algebraical border rank of t, and \(\underset \tilde{} R(t)\) is topological border rank of t, then, the main result is the following Theorem. For \(t\in k^ n\otimes k^ m\otimes k^ l\) the following statements are equivalent: (i) \(\underset \tilde{} R(t)\leq r\), (ii) Ṟ(t)\(\leq r\), (iii) \(R_ h(t)\leq r\), where \[ h=(\frac{(n+m+1-3)!}{[(n-1)!(m-1)!(l- 1)!]})r\leq 3^{(n+m+1-3)r} \] if k is algebraically closed, and \(h=3((n+m+1-2)r+10)^{2(n+m+1-2)r-1}-1\) if k is real closed.