{
  "annotation_id": "78919f30-622f-4f6a-a6dc-94ffcae3e127",
  "date_created": "2026-02-17T05:53:12.325000Z",
  "date_modified": "2026-02-17T05:53:12.325000Z",
  "file_hash": "e19a943403663401256ef328b8159fc969147db5b94894941125db529a5f74cb",
  "private": false,
  "record": {
    "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"
    ],
    "categories": [
      "math.GM"
    ],
    "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",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e1a039bb-3f1e-44d1-a2e3-2625df02ab07",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}