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

Sanda Bujačić Babić; Ana Jurasić

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

Number Theory 2026-07-30 v1

Authors: Sanda Bujačić Babić , Ana Jurasić

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 $\mathbb{C}[X]$ .

Keywords

Diophantine quadruple regularity polynomial ring Gaussian rational Diophantine m-tuples

Cite

@article{arxiv.2607.28152,
  title  = {Regularity of Diophantine quadruples over $\mathbb{Q}(i)[X]$},
  author = {Sanda Bujačić Babić and Ana Jurasić},
  journal= {arXiv preprint arXiv:2607.28152},
  year   = {2026}
}