## The Hyperdeterminant Book

October 3, 2008

There is currently only one book that covers the subject matter of hyperdeterminants in any detail. “Discriminants, Resultants and Multidimensional Determinants” by I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky.

Gelfand at 95 years old is now in Rutgers University USA. He is highly regarded as a great achievers in diverse areas of algebra and geometry and has been awarded many honours including the Order of Lenin and the Wolf Prize. His younger co-authors Kapranov and Zelevinsky have also been his collobarators for many years.

Their interest in hyperdeterimants came from research into generalized hyperegometric functions when they found that the leading terms were the inverses of a generalised type of discriminant. They then relaised that they had in fact rediscovered the hyperdeterminants first studied by Cayley in 1845. When they read Cayley’s original apprach they were astonished at how many modern concepts he had discovered that had then been neglected or forgotten. Soon they came to the conclusion that hyperdeterminants and related dicriminants deserved a book in their own right without reference to the hypergeometric functions which had led to them.

The book is written in the language of algebraic geometry covering such areas as Chow Varieties, Toric Varieties and the authors generalization of discriminants which they call A-Disciminants. Other mathematicians may call them GKZ-discriminants in honour of the authors. Finally in Chapter 14 they arrive at the subject of hyperdeterminants which are then just special cases of their discriminants. The general theory can be applied to derive results about their degree and other properties.

Of special interest in the book is the relationship between the Newton Polytope of the hyperdeterminant and the combinatorial problem of counting the number of distinct ways of triangulating a cube in a given number of dimensions. The introduction provides some insight into the wonderment that this connection brought them. They note that the coefficients that appear in discriminants are often products of factors of the form VVwhere V is the volume of a simplex in a particular triangulation. The logs of these terms then “bring to mind the entropy of a probability distribution”. This feature has appeared elsewhere in related work and begs for a deeper explanation. To a physicist it might suggest a deep connection with the dynamical triangulation approaches to quantum gravity.

Despite its depth the book is still just an introduction to hyperdeteriminants and does not cover recently discovered connections with areas such as number theory and string theory, but it is an excellent reference point for anyone who wants to help push this subject forward.

In its hardback form it used to be a bit too expensive for most peoples personal library but sufficient interest has now seen the release of a paperback version at a very reasonable price