Hiperdeterminantlar, dolanık durumlar ve değişmezler teorisi

In [1] and [2], A. Klyachko connects quantum entanglement and invariant theory so that entangled state of a quantum system can be explained by invariants of the system. After representing states in multidimensional matrices, this relation turns into finding multidimensional matrix invariants so called hyperdeterminants. Here we provide a necessary and suffi cient condition for existence of a hyperdeterminant of a multidimensional matrix of an arbitrary format. The answer is given in terms of the so called castling transform that relates hyperdeterminants of diff erent formats. Among castling equivalent formats there is a unique castling reduced one, that has minimal number of entries. We prove the following theorem: "Multidimensional matrices of a given format admit a non-constant hyperdeterminant if and only if logarithm of dimensions of the castling reduced format satisfy polygonal inequalities."