Polynomial $D(4)$-quadruples over Gaussian Integers

TL;DR

This paper proves that all polynomial D(4)-quadruples over Gaussian integers are regular, satisfying a specific polynomial equation, thus classifying their structure in the Gaussian integer polynomial ring.

Contribution

It establishes that every D(4)-quadruple over Gaussian integers is regular, confirming a conjecture about their structure in polynomial rings over Gaussian integers.

Findings

All D(4)-quadruples are regular.

The key equation $(a+b-c-d)^2=(ab+4)(cd+4)$ holds universally.

Classification of D(4)-quadruples over Gaussian integers achieved.

Abstract

A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in $\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\mathbb{Z}[i][X]$. In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$.