Non-Existence of Linear-Quartic Factorization for the Second Cuboid Quintic
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918282807738368 |
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| author | Asiryan, Valery |
| author_facet | Asiryan, Valery |
| contents | Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] We show that for every rational $s>0$ with $s\neq 1$ the equation $P_s(x)=0$ has no rational solutions. Equivalently, $P_s$ admits no $1+4$ factorization over $\mathbb{Q}$.
The proof uses an explicit quotient by the inversion involution $(s,y)\mapsto(1/s,1/y)$ and reduces the rational-root problem for $P_s$ to rational points on the fixed genus-$2$ hyperelliptic curve \[ C:\quad w^2=t^5+21t^4+26t^3+10t^2+5t+1=(t+1)(t^4+20t^3+6t^2+4t+1). \] Using Magma and Chabauty's method on the Jacobian of $C$, we compute $C(\mathbb{Q})$ exactly and conclude that the only parameter value producing a rational root is the excluded case $s=1$ (equivalently $p=q$).
As a consequence, for coprime $p\neq q>0$ the polynomial $Q_{p,q}(t)$ has no rational roots (hence no linear factor over $\mathbb{Q}$, and in particular no linear factor over $\mathbb{Z}$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Existence of Linear-Quartic Factorization for the Second Cuboid Quintic Asiryan, Valery General Mathematics 11D41, 11G30, 14H25, 12E05, 11Y16 Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] We show that for every rational $s>0$ with $s\neq 1$ the equation $P_s(x)=0$ has no rational solutions. Equivalently, $P_s$ admits no $1+4$ factorization over $\mathbb{Q}$. The proof uses an explicit quotient by the inversion involution $(s,y)\mapsto(1/s,1/y)$ and reduces the rational-root problem for $P_s$ to rational points on the fixed genus-$2$ hyperelliptic curve \[ C:\quad w^2=t^5+21t^4+26t^3+10t^2+5t+1=(t+1)(t^4+20t^3+6t^2+4t+1). \] Using Magma and Chabauty's method on the Jacobian of $C$, we compute $C(\mathbb{Q})$ exactly and conclude that the only parameter value producing a rational root is the excluded case $s=1$ (equivalently $p=q$). As a consequence, for coprime $p\neq q>0$ the polynomial $Q_{p,q}(t)$ has no rational roots (hence no linear factor over $\mathbb{Q}$, and in particular no linear factor over $\mathbb{Z}$). |
| title | Non-Existence of Linear-Quartic Factorization for the Second Cuboid Quintic |
| topic | General Mathematics 11D41, 11G30, 14H25, 12E05, 11Y16 |
| url | https://arxiv.org/abs/2601.04241 |