{
  "abstract": "We study simultaneous popular differences for polynomial progressions and linear progressions over finite fields $\\mathbb F_p$. For certain families of polynomial progressions of algebraic complexity $1$, such as $(x, x+y, x+2y)$ and $(x, x+y^3, x+2y^3)$, we prove the existence of $y \\neq 0$ which is popular simultaneously for both progressions. This extends previous results by Conlon, Dong and Hong who considered complexity $0$ progressions. On the other hand, we construct counterexamples showing that simultaneous popular differences are not guaranteed to exist for $(x, x+y, x+2y)$ and $(x, x+y^2, x+2y^2)$, or for $(x, x+y, x+2y)$ and $(x, x+y, x+3y)$, or for $(x, x+y, x+2y)$ and $(x,x+y,x+2y,x+3y)$, answering questions posed by Conlon, Dong and Hong.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28226",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuancheng Shao"
    },
    {
      "name": "Weiwen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvf51qkyqb4b4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28226",
  "fulltext_url": null,
  "id": "dvf51qkyqb4b4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28226",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223085787",
  "pdf_url": "https://arxiv.org/pdf/2610.28226",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvf51qkyqb4b4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28226",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28226"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Simultaneous popular differences for polynomial and linear progressions in finite fields",
  "title_text": "Simultaneous popular differences for polynomial and linear progressions in finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/dvf51qkyqb4b4",
  "venue": "arXiv",
  "versions": []
}