The Minimal Euclidean Function on the Gaussian Integers

TL;DR

This paper introduces the first explicit minimal Euclidean function for Gaussian integers, enabling efficient computation of shortest $(1+i)$-ary expansions in the Gaussian integer domain.

Contribution

It provides the first explicit minimal Euclidean function for a non-trivial number field, specifically for Gaussian integers, and presents an algorithm for minimal expansions.

Findings

Explicit minimal Euclidean function for Gaussian integers derived

Algorithm for minimal $(1+i)$-ary expansions developed

Elementary methods used to solve complex number field problems

Abstract

In 1949, Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$. He showed that when $R = \mathbb{Z}$, the minimal function is $\phi_{\mathbb{Z}}(x) = \lfloor \log_2 |x| \rfloor$. For over seventy years, $\phi_{\mathbb{Z}}$ has been the only example of an explictly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $\phi_{\mathbb{Z}[i]}$, which computes the length of the shortest possible $(1+i)$-ary expansion of any Gaussian integer. We also present an algorithm that uses $\phi_{\mathbb{Z}[i]}$ to compute minimal $(1+i)$-ary expansions of Gaussian integers. We solve these problems using only elementary methods.