{
  "abstract": "For any $m, s \\in \\mathbb{N}$, we study the number $N_{m\\times s,q}(κ, β)$ of solutions $(x_1,\\ldots,x_s) \\in (\\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \\cdots + X_{s}^{k_i}= β_i, \\qquad (1\\le i \\le m), $$ with $κ=(k_1,\\ldots,k_m) \\in \\mathbb{N}^m$ and $β=(β_1,\\ldots,β_m) \\in (\\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of \\textit{diagonal} GP-graphs $Γ(κ,q)$. This is a new family of graphs that we introduce here, $Γ(κ,q)$, with $κ= (k_1,\\ldots,k_m) \\in \\mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \\in R_κ = \\{ (x^{k_1},\\ldots,x^{k_m}) : x \\in \\mathbb{F}_{q}^*\\}$. In particular, we give three different expressions for $N_{m\\times s,q}(κ, β)$: one in terms of walks, another in terms of adjacency matrices of $Γ(κ,q)$ and the last one in terms of the spectrum of $Γ(κ,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\\times s,q}(κ_\\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\\ell_i}+1} + \\cdots + X_s^{q^{\\ell_i}+1} = 0 \\qquad (1\\le i \\le m),$$ with $κ_\\ell=(\\ell_1,\\ldots,\\ell_m)=(1,3,\\ldots,2m-1)$ and $m\\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \\in \\mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \\in \\mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\\le s \\le 5$, we give explicit expressions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19507",
  "author_total": 2,
  "authors": [
    {
      "name": "Ricardo A. Podestá"
    },
    {
      "name": "Denis E. Videla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7a7ssseag6vhx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19507",
  "fulltext_url": null,
  "id": "7a7ssseag6vhx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19507",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7203874003",
  "pdf_url": "https://arxiv.org/pdf/2608.19507",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7a7ssseag6vhx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.19507",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19507"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases",
  "title_text": "On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7a7ssseag6vhx",
  "venue": "arXiv",
  "versions": []
}