Four-square polynomial triples: quadratic classification and equal-degree obstructions

Abstract

We study triples of distinct nonconstant polynomials a,b,c over ℝ, ℚ or ℤ for which ab+1, ac+1, bc+1, abc+1 are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in ℝ[X], the one arising from a recent construction of Jurasić, and that none exists in ℚ[X] or ℤ[X]. Moreover, for arbitrary nonconstant entries over ℝ or ℚ, we prove that the degree pattern $(2n,2m,2l)$ with 1≤ n≤ m≤ l satisfies either l=m or l>n+m. For the case l=m, we derive factorization systems and reducibility restrictions. In the equal-degree case over ℚ, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in ℚ[X] the fourth expression abc+1 is never a square when the parameter is constant or the first entry is a square.

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BibTeX RIS

BibTeX

@misc{bokun2026foursquare,
  title = {{Four-square polynomial triples: quadratic classification and equal-degree obstructions}},
  author = {Mirela Jukić Bokun and Ana Jurasić},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05565},
  eprint = {2610.05565},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05565},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Four-square polynomial triples: quadratic classification and equal-degree obstructions
AU  - Mirela Jukić Bokun
AU  - Ana Jurasić
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05565
UR  - https://arxiv.org/abs/2610.05565
AB  - We study triples of distinct nonconstant polynomials a,b,c over ℝ, ℚ or ℤ for which ab+1, ac+1, bc+1, abc+1 are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in ℝ[X], the one arising from a recent construction of Jurasić, and that none exists in ℚ[X] or ℤ[X]. Moreover, for arbitrary nonconstant entries over ℝ or ℚ, we prove that the degree pattern $(2n,2m,2l)$ with 1≤ n≤ m≤ l satisfies either l=m or l>n+m. For the case l=m, we derive factorization systems and reducibility restrictions. In the equal-degree case over ℚ, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in ℚ[X] the fourth expression abc+1 is never a square when the parameter is constant or the first entry is a square.
ER  -

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