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

Sanda Bujačić Babić; Ana Jurasić

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arXiv · 2607.28152

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

Sanda Bujačić Babić·Ana Jurasić

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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 $\mathbb{C}[X]$.

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Sanda Bujačić Babić, Ana Jurasić. 2026-07-30. Regularity of Diophantine quadruples over $\mathbb{Q}(i)[X]$. https://arxiv.org/abs/2607.28152

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