Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ via a Rank-Zero Elliptic Curve
Valery Asiryan
TL;DR
Problem: prove that $P_{a,u}(t)$ is irreducible in $\mathbb{Z}[t]$ for coprime integers $a\neq u>0$. Approach: factor over $K=\mathbb{Q}(\sqrt2)$ as two coprime quartics $H_-(t)H_+(t)$, reduce potential $K$-factorizations to rational points on the genus-one quartic $\mathcal{C}$ via $\tau=(au/\Delta)^2$, and analyze the Jacobian of $\mathcal{C}$ by modeling it with an elliptic curve $E$ of rank $0$ and torsion $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. Result: the only rational points on $\mathcal{C}$ give $\tau\in\{0,1/4\}$, which are incompatible with coprime $a,u$, hence $P_{a,u}(t)$ is irreducible in $\mathbb{Q}[t]$ and in $\mathbb{Z}[t]$ by Gauss. Significance: demonstrates an arithmetic irreducibility proof for a family tied to the perfect cuboid problem using genus-one curves and elliptic techniques, with Magma confirming rank and torsion.
Abstract
For coprime integers $a\neq u>0$ put $Δ:=u^2-a^2\neq0$ and $A_0:=a^2u^2$. Consider \[ P_{a,u}(t)=t^8+6Δ\,t^6+(Δ^2-2A_0)\,t^4-6ΔA_0\,t^2+A_0^2\in\mathbb{Z}[t]. \] We prove $P_{a,u}(t)$ is irreducible over $\mathbb{Z}$. The argument is: (1) over $K=\mathbb{Q}(\sqrt2)$, $P_{a,u}$ splits as $H_-H_+$ with coprime conjugate quartics $H_\pm$; (2) any hypothetical $K$--factorization of $H_\pm$ forces a rational point on a fixed genus--one quartic $\mathcal{C}: v^2=16y^4+136y^2+1$ with the structural constraint $τ=y^2=(au/Δ)^2$; (3) the Jacobian of $\mathcal{C}$ admits a Weierstrass model $E/\mathbb{Q}$ on which a Magma computation certifies $\mathrm{rank}\,E(\mathbb{Q})=0$ and $E(\mathbb{Q})_{\mathrm{tors}}\cong \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$; (4) the only rational $τ$ arising from $\mathcal{C}(\mathbb{Q})$ are $τ\in\{0,1/4\}$, incompatible with coprime integers $a\neq u>0$. Therefore $H_\pm$ are irreducible in $K[t]$, whence $P_{a,u}$ is irreducible in $\mathbb{Q}[t]$ and in $\mathbb{Z}[t]$ by Gauss.