Irreducibility of the Cuboid Polynomial Qp,q(t) via the Rank-One Chabauty-Coleman Method
Abstract
Let Qp,q(t)∈ℤ[t] be Sharipov's even monic degree-$10$ second cuboid polynomial, with coprime positive integers p≠ q. We prove that Qp,q(t) is irreducible over ℚ, and hence over ℤ. More generally, the normalized associated quintic Ps(x) and its composition Ps(t2) are irreducible over ℚ for every s∈ℚ∖{0,±1}. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-$2$ curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime $3$. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are $3$-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative real root of the irreducible quintic then supplies the additional degree argument required for irreducibility of the degree-$10$ polynomial. The rank computation and all exact arithmetic certificates are specified explicitly for reproduction.
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BibTeX
@misc{asiryan2026irreducibility,
title = {{Irreducibility of the Cuboid Polynomial $Q_p,q(t)$ via the Rank-One Chabauty-Coleman Method}},
author = {Valery Asiryan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25319},
eprint = {2610.25319},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25319},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irreducibility of the Cuboid Polynomial Q_p,q(t) via the Rank-One Chabauty-Coleman Method
AU - Valery Asiryan
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25319
UR - https://arxiv.org/abs/2610.25319
AB - Let Q_p,q(t)∈ℤ[t] be Sharipov's even monic degree-$10$ second cuboid polynomial, with coprime positive integers p≠ q. We prove that Q_p,q(t) is irreducible over ℚ, and hence over ℤ. More generally, the normalized associated quintic P_s(x) and its composition P_s(t^2) are irreducible over ℚ for every s∈ℚ∖{0,±1}. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-$2$ curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime $3$. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are $3$-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative real root of the irreducible quintic then supplies the additional degree argument required for irreducibility of the degree-$10$ polynomial. The rank computation and all exact arithmetic certificates are specified explicitly for reproduction.
ER -