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Two-Sided Dimension Bounds for the Peterson Hit Problem via Projections and Matrix Minors — arXiv:2608.02623

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.02623
License (as archived): CC0-1.0
Archived: 2026-10-31T12:55:56.232Z
SHA-256 of the archived text: d68810011eb5c2c64c1d85f7de91215d692646f192d3865374faf4a26abd9aca
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How to cite

Two-Sided Dimension Bounds for the Peterson Hit Problem via Projections and Matrix Minors — arXiv:2608.02623 — arXiv (en), https://arxiv.org/abs/2608.02623 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cb8755dbee85 (Metraipolis archive dc_cb8755dbee85, sha256 d68810011eb5c2c64c1d85f7de91215d692646f192d3865374faf4a26abd9aca)

BibTeX

@misc{dccb8755dbee85,
  title = {Two-Sided Dimension Bounds for the Peterson Hit Problem via Projections and Matrix Minors — arXiv:2608.02623},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.02623},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cb8755dbee85; sha256 d68810011eb5c2c64c1d85f7de91215d692646f192d3865374faf4a26abd9aca},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.02623. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $\mathcal P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra over the prime field $\mathbb F_2$, viewed as an unstable module over the mod-$2$ Steenrod algebra $\mathcal A$. The well-known Peterson hit problem asks for a minimal set of generators for the $\mathcal A$-module $\mathcal P_k$. This is equivalent to determining the dimension of the cohit space $(Q\mathcal P_k)_d=(\mathcal P_k/\mathcal A^{+}\mathcal P_k)_d$, where $\mathcal A^{+}$ denotes the augmentation ideal of $\mathcal A$, for every $k\geq1$ and positive degree $d$. Although solved in every degree for at most four variables, it remains a difficult open problem in general. Furthermore, given the limitations of current tools, explicitly determining the dimension of $(Q\mathcal P_k)_d$ in the general case appears out of reach. Motivated by these limitations, we establish explicit upper and lower bounds for this dimension for arbitrary positive integers $k$ and $d.$ Our method combines binary combinatorics, linear algebra, and graph and simplicial structures associated with the generating Steenrod squares. We characterize zero rows, count zero columns, and refine rank estimates using Adem relations. Minors and zero rows of the resulting smaller matrix yield further two-sided cohit bounds without determining a complete basis or computing the full hit rank.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-05 in math.CO).
Categorie arXiv: Algebraic Topology (math.AT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.02623 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-31T12:42:47.790Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-05?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).