Nondegenerate $2 \times k \times (k+1)$ Hypermatrices

TL;DR

This paper extends Gaussian elimination to analyze the topology and algebraic properties of nondegenerate 2 x k x (k+1) hypermatrices over topological fields, providing new computational and theoretical insights.

Contribution

It introduces a novel group action on hypermatrices, enabling the determination of homotopy groups, counting over finite fields, and efficient hyperdeterminant computation.

Findings

Determined homotopy groups of $M_k(\mathbb{C})$

Counted elements of $M_k(\mathbb{F}_q)$

Developed $O(k^4)$ hyperdeterminant algorithm

Abstract

We construct an extension of Gaussian elimination to show that if $\mathbb{F}$ is a topological field, then there is a transitive, free, and continuous action of a natural quotient of $GL_k(\mathbb{F}) \times GL_{k+1}(\mathbb{F})$ on the set $M_k(\mathbb{F})$ of $2 \times k \times (k+1)$ hypermatrices over $\mathbb{F}$ with nonzero hyperdeterminant. We use this action to answer a number of questions including determining the homotopy groups of $M_k(\mathbb{C})$, counting elements of $M_k(\mathbb{F}_q)$ (generalizing an unpublished result of Lewis and Sam), and computing hyperdeterminants for $2 \times k \times (k+1)$ hypermatrices in $O(k^4)$ time, which we use to compute explicit formulas in some special cases.