{
  "abstract": "Bachmann and van Ittersum conjectured that every polynomial $f\\in\\mathbb{Q}[x,y,z]$ satisfying $$ f(x,y,z)=f(y,z,x),\\qquad f(x,y,z)+f(-y,x-y,z)+f(y-x,-x,z)=0 $$ is a sum of polynomials in two variables. We prove this conjecture over every field of characteristic zero and determine all homogeneous solutions. Moreover, restriction to $z=0$ identifies the degree $n$ solution space with the space of binary period polynomials of degree $n$ normalized to vanish on both coordinate axes. The solution corresponding to such a polynomial $q$ is $$ f(x,y,z)=q(x,y)+q(y,z)+q(z,x). $$ There are no nonzero homogeneous solutions of odd degree. For even degrees $n$, the symmetric and alternating solution spaces are each isomorphic over $\\mathbb{C}$ to the space of weight $n+2$ cusp forms on $SL_2(\\mathbb{Z})$. The proof connects the polynomial relations with the cohomology of arithmetic groups through the rational Tits building for $GL_3$. Cusp forms enter both the description of the solutions and the proof that there are no others. The main theorems and the essential argument are formalized unconditionally in Lean~4 by AxiomProver.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19309",
  "author_total": 2,
  "authors": [
    {
      "name": "Vasily Ilin"
    },
    {
      "name": "Ken Ono"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rdp7htbgaprk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19309",
  "fulltext_url": null,
  "id": "9rdp7htbgaprk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19309",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7222184502",
  "pdf_url": "https://arxiv.org/pdf/2610.19309",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rdp7htbgaprk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19309",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19309"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ternary polynomial relations and periods of cusp forms",
  "title_text": "Ternary polynomial relations and periods of cusp forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/9rdp7htbgaprk",
  "venue": "arXiv",
  "versions": []
}