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

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 \(D(4)\)-q...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Trebješanin, Marija Bliznac, Babić, Sanda Bujačić
Format: Article
Language:English
Subjects:
Polynomials
Online Access:Get full text
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Summary: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 \(\mathbb{Z}[i][X]\).
ISSN:2331-8422
DOI:10.48550/arxiv.2210.10575