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When is there a multipartite maximum entangled state?

For a multipartite quantum system of the dimension d1 ⊗ d2 ⊗ ... ⊗ dn, where d1 ≥d2 ≥ ... ≥ dn ≥ 2, is there an entangled state maximum in the sense that all otherstates in the system can be obtained from the state through local quantum operationsand classical communications (LOCC)? When d1 ≥ πi=2ndi such state exists. We showthat this condition is also necessary. Our proof, somewhat surprisingly, uses results fromalgebraic complexity theory.