Hyperdeterminants as integrable discrete systems

TL;DR

This paper explores hyperdeterminants, proving the integrability of a specific nonlinear difference equation derived from the 2x2x2 hyperdeterminant, and hypothesizes about the integrability of hyperdeterminants of larger sizes.

Contribution

It establishes the integrability of a difference equation from the 2x2x2 hyperdeterminant and proposes a hypothesis on the integrability of hyperdeterminants of any size.

Findings

Proves integrability (4d-consistency) of the 2x2x2 hyperdeterminant-based difference equation.

Hypothesizes that hyperdeterminants of larger sizes are also integrable.

Shows that the hypothesis fails for the 2x2x2x2 hyperdeterminant.

Abstract

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability (understood as 4d-consistency) of a nonlinear difference equation defined by the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the 2x2x2x2-hyperdeterminant.