Diophantine quadruples in \(\mathbb{Z}[i][X]\)
``A set consisting of \(m\) positive integers such that the product of any two of its distinct elements increased by \(1\) is a perfect square is called a Diophantine \(m\)-tuple. There is long history of finding such sets. One of the questions of interest, which various mathematicians try to solve, is how large those sets can be. Very recently, \textit{B. He} et al. [Trans. Am. Math. Soc. 371, No. 9, 6665--6709 (2019; Zbl 1430.11044)] proved the folklore conjecture that there cannot be \(5\) elements in Diophantine \(m\)-tuple, i.e. \(m < 5\).'' \par ``There are also many generalizations of the original problem, for example we can add a fixed integer \(n\) instead of \(1\) or consider the problem over domains other than \(\mathbb{Z}\) or \(\mathbb{Q}\).'' \par ``Definition. Let \(m \geq 2\) and let \(R\) be a commutative ring with unity. Let \(n \in R\) be a non-zero element and let \(\{a_1,\dots, a_m\}\) be a set of \(m\) distinct non-zero elements from \(R\) such that \(a_ia_j + n\) is a square of an element of \(R\) for \(1 \leq i < j \leq m\). The set \(\{a_1,\dots, a_m\}\) is called a Diophantine \(m\)-tuple with the property \(D(n)\) or simply a \(D(n)-m\)-tuple in \(R\).'' \par The authors consider a polynomial variant of the problem, with \(R = \mathbb{Z}[i][X]\) and \(n = 1\). The authors prove that every Diophantine quadruple in \(\mathbb{Z}[i][X]\) is regular (Theorem 1.5). More precisely, they prove that if \(\{a,b,c,d\}\) is a set of non-zero polynomials from \(\mathbb{Z}[i][X]\), not all constant, such that the product of any of its distinct elements increased by \(1\) is a square of a polynomial from \(\mathbb{Z}[i][X]\), then \((a+b-c-d)^2 = 4(ab+1)(cd+1)\). \par ``Since we do not have the relation ``\(<\)'' between the elements of \(\mathbb{Z}[i][X]\), we use the relation ``\(\leq\)'' between their degrees. In order to prove Theorem 1.5, we partially follow the strategies used in \textit{A. Dujella} and \textit{C. Fuchs} [J. Number Theory 106, No. 2, 326--344 (2004; Zbl 1047.11024)] for \(\mathbb{Z}[X]\), in \textit{A. Filipin} and \textit{A. Jurasić} [Glas. Mat., III. Ser. 54, No. 1, 21--52 (2019; Zbl 1455.11047)] for \(\mathbb{R}[X]\) and in \textit{A. Dujella} and \textit{A. Jurasić} [Int. J. Number Theory 6, No. 7, 1449--1471 (2010; Zbl 1255.11011)] for \(\mathbb{C}[X]\), but we also introduce a few new ideas.''
scientific article; zbMATH DE number 7377563
scientific article