Four-square polynomial triples: quadratic classification and equal-degree obstructions
We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ 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 $\mathbb{R}[X]$, the one arising from a recent construction of Jurasić, and that none exists in $\mathbb{Q}[X]$ or $\mathbb{Z}[X]$. Moreover, for arbitrary nonconstant entries over $\mathbb{R}$ or $\mathbb{Q}$, we prove that the degree pattern $(2n,2m,2l)$ with $1\le n\le m\le l$ 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 $\mathbb{Q}$, 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 $\mathbb{Q}[X]$ the fourth expression $abc+1$ is never a square when the parameter is constant or the first entry is a square.