s-Hankel hypermatrices and 2 × 2 determinantal ideals

We introduce the concept of an s-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an s-Hankel hypermatrix: the ideal I〈s,t〉 generated by certain 2 × 2 slice minors, and the ideal Ĩ〈s,t〉 generated by certain 2 × 2 generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of I〈s,t〉, and provide the explicit list of minimal primes for large values of s. Finally we give some geometrical interpretations and generalize a theorem of Watanabe.