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Given a vector space V over a field whose characteristic is coprime with d!, let us decompose the vector space of multilinear forms V^* (d) V^*= _ W_ (X, ) according to the different partitions of d, i. e. the different representations of Sd. In this paper we first give a decomposition W (₃-₁, ₁) (V, ) =₈=₁^d-1W (₃-₁, ₁) ⁱ (V, ). We finally prove the vanishing of the hyperdeterminant of any F ( (₃), (₃-₁, ₁) ) W (₃-₁, ₁) ⁱ (V, ). This improves the result in 10 and 1, where the same result was proved without this new last summand.
Arrondo et al. (Tue,) studied this question.