Regularity of Diophantine quadruples over $\mathbb{Q}(i)[X]$

Abstract

We prove that every Diophantine quadruple $\{a,b,c,d\}$ over $\mathbb{Q}(i)[X]$ that contains at least one non-constant polynomial is regular; that is, $$(a+b-c-d{)}^2=4(ab+1)(cd+1).$$ This result is consistent with the corresponding results over $\mathbb{Z}[i][X]$ and $\mathbb{R}[X]$, but contrasts with the situation over $\mathbb{C}[X]$.