Determinants of Steiner Distance Hypermatrices

Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-k Steiner distance hypermatrices of trees on n vertices. We show that they can be nearly diagonalized as k-forms, generalizing a result of Graham-Lov\'{a}sz, implying a tensor version of ``conditional negative definiteness'', providing new proofs of previous results of the authors and Tauscheck, and resolving the conjecture that these hyperdeterminants depend only on k and n -- as Graham-Pollak showed for $k=2$. We conclude with some open questions.