Abundance of Matrices In Gaussian Integers

TL;DR

This paper extends the concept of abundance of matrices from rational entries to matrices over Gaussian integers, overcoming challenges posed by the complex field's lack of order.

Contribution

It proves the abundance property for matrices with Gaussian integer entries, a significant generalization from previous rational-based results.

Findings

Established abundance for matrices over Gaussian integers

Overcame the obstacle of no linear order in complex numbers

Extended previous rational matrix results to complex integer matrices

Abstract

In [HLS], N. Hindman, I. Leader and D. Strauss proved the abundance for a matrix with rational entries. In this paper we proved it for the ring of Gaussian integers. We showed the result when the matrix is taken with entries from \mathbb{Q}\left[i\right]. The main obstacle is in the field of complex numbers, no linear order relation exists. We overcome that in a tactful way.