Excluding 54 of the 76 Nontrivial Factorization Patterns of the Cuboid Polynomial Pa,b,u(t)
Abstract
Let Pa,b,u(t) be Sharipov's even monic degree-$12$ third cuboid polynomial, with positive pairwise distinct rational parameters. We prove a uniform normal form for its complete factorization over ℚ. Every factorization consists of at most two even quadratic factors, even irreducible factors of higher degree, and pairs of irreducible factors exchanged by t↦−t. After dividing all degrees by the contribution of this symmetry, the normal form satisfies one degree equation, one sign equation, and an obligatory-factor alternative: an even irreducible factor of degree at least six, or an even irreducible quartic with positive rational-square constant term. The arithmetic input is a uniform construction over ℚ2, using Newton polygons, Hensel lifting, and a norm obstruction from ℚ2(i). Valuations at all finite primes constrain rational factor constants to signed squares, while the real root equation gives exactly two simple negative roots of the auxiliary sextic. The normal form leaves exactly $22$ of the $76$ nontrivial partitions of $12$, thereby excluding $54$ complete irreducible degree patterns by one argument. Of these exclusions, $37$ follow from evenness alone and $17$ from the additional arithmetic restrictions. All local cases are proved explicitly, and an exact verification script accompanies the algebraic identities and exhaustive partition enumeration. The remaining patterns are necessary possibilities; their realization is not asserted, and full irreducibility remains open.
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BibTeX
@misc{asiryan2026excluding,
title = {{Excluding 54 of the 76 Nontrivial Factorization Patterns of the Cuboid Polynomial $P_a,b,u(t)$}},
author = {Valery Asiryan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10012},
eprint = {2611.10012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10012},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Excluding 54 of the 76 Nontrivial Factorization Patterns of the Cuboid Polynomial P_a,b,u(t)
AU - Valery Asiryan
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.10012
UR - https://arxiv.org/abs/2611.10012
AB - Let P_a,b,u(t) be Sharipov's even monic degree-$12$ third cuboid polynomial, with positive pairwise distinct rational parameters. We prove a uniform normal form for its complete factorization over ℚ. Every factorization consists of at most two even quadratic factors, even irreducible factors of higher degree, and pairs of irreducible factors exchanged by t↦−t. After dividing all degrees by the contribution of this symmetry, the normal form satisfies one degree equation, one sign equation, and an obligatory-factor alternative: an even irreducible factor of degree at least six, or an even irreducible quartic with positive rational-square constant term. The arithmetic input is a uniform construction over ℚ_2, using Newton polygons, Hensel lifting, and a norm obstruction from ℚ_2(i). Valuations at all finite primes constrain rational factor constants to signed squares, while the real root equation gives exactly two simple negative roots of the auxiliary sextic. The normal form leaves exactly $22$ of the $76$ nontrivial partitions of $12$, thereby excluding $54$ complete irreducible degree patterns by one argument. Of these exclusions, $37$ follow from evenness alone and $17$ from the additional arithmetic restrictions. All local cases are proved explicitly, and an exact verification script accompanies the algebraic identities and exhaustive partition enumeration. The remaining patterns are necessary possibilities; their realization is not asserted, and full irreducibility remains open.
ER -