dorsal/arxiv
View SchemaComputational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic
| Authors | Valery Asiryan, Randall L. Rathbun |
|---|---|
| Categories | |
| ArXiv ID | 2601.07899vv2 |
| URL | https://arxiv.org/abs/2601.07899 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
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 \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] Assuming a quadratic divisor $x^{2}+ax+b$ with $a,b\in\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder \[ R(x)=R_{1}(s,a,b)\,x+R_{0}(s,a,b). \] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\in\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\pm 1$ (hence only $s=1$ in the cuboid domain $s>0$). Let $\overline{C}\subset\mathbb{P}^{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\overline{C}$. With bound $H=10^{7}$, the search returns $8$ rational points, whose affine part has $s\in\{-1,0,1\}$. In particular, no affine rational point with $s>0$ and $s\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s>0$, $s\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\overline{C}$.
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"abstract": "Let $Q_{p,q}(t)\\in\\mathbb{Z}[t]$ be Sharipov\u0027s even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\\neq q\u003e0$. 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}. \\] Assuming a quadratic divisor $x^{2}+ax+b$ with $a,b\\in\\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder \\[ R(x)=R_{1}(s,a,b)\\,x+R_{0}(s,a,b). \\] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\\in\\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\\pm 1$ (hence only $s=1$ in the cuboid domain $s\u003e0$). Let $\\overline{C}\\subset\\mathbb{P}^{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\\overline{C}$. With bound $H=10^{7}$, the search returns $8$ rational points, whose affine part has $s\\in\\{-1,0,1\\}$. In particular, no affine rational point with $s\u003e0$ and $s\\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s\u003e0$, $s\\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\\overline{C}$.",
"arxiv_id": "2601.07899",
"authors": [
"Valery Asiryan",
"Randall L. Rathbun"
],
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"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Computational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic",
"url": "https://arxiv.org/abs/2601.07899",
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