On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909839487139840 |
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| author | Asiryan, Valery |
| author_facet | Asiryan, Valery |
| contents | In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2{+}6$, the patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ regroup trivially to $2{+}6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ Asiryan, Valery General Mathematics 12E05, 11D72, 11S05, 11Y05 In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2{+}6$, the patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ regroup trivially to $2{+}6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$. |
| title | On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ |
| topic | General Mathematics 12E05, 11D72, 11S05, 11Y05 |
| url | https://arxiv.org/abs/2510.07643 |