Enhanced Algorithm for Computing Cayley’s First Hyperdeterminant

Combinatorial hyperdeterminant DET – is the homogeneous polynomial in the entries of a hypermatrix of even number of indices, which is also a unique SL-invariant of minimal degree. It was first studied by Cayley in the middle of 19-th century. Given its fundamental nature, the computation of this polynomial is an important task. For fixed d and a cubical hypermatrix X of length n Barvinok introduced an algorithm of computing hyperdeterminant in 0(2 nd n d-1 ) . Since the problem of deciding whether for the given hypermatrix X the hyperdeterminant DET(X) is equal to zero is NP-hard, it is essential to develop efficient algorithm for computing hyperdeterminant, as the size of problem grows exponentially. We provide enhanced algorithm of computing hyperdeterminant that requires 0(2 n(d-1) n d-1 ) arithmetic operations.