发表机构
FPT University; Department of Mathematics and Statistics, Quy Nhon University; Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（FPT大学; 平定大学数学与统计系; 保加利亚科学院数学与信息学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对$k \boldsymbol{\textit{≥}} 5$的彼得森击中问题，构建稀疏矩阵并推导认证矩阵子式，得到对矩阵秩的双侧组合界，为该问题提供了可计算的通用离散认证。
AI 中文摘要
彼得森击中问题旨在寻找多项式代数$\boldsymbol{\textit{P}}_k=\boldsymbol{\textit{F}}_2[x_1,\boldsymbol{\textit{…}},x_k]$作为模2 Steenrod代数模的极小生成元集合。该问题在$k \boldsymbol{\textit{≤}} 4$时已完全解决，但$k \boldsymbol{\textit{≥}} 5$的无限制问题仍悬而未决，因基元的组合爆炸使精确算法计算难以实现。为绕过完全高斯消元，我们利用Cartan公式和Lucas定理构建由稀疏矩阵驱动的$d$次击中空间，将研究重点转向认证矩阵子式的构造。我们首先证明严格尖单项式精确刻画零行，建立坐标级零化子的严格结构限制；为从上方界定矩阵秩（即余击中下界），推导精确零列公式，该公式被$\boldsymbol{\textit{Sq}}^1$层的精确同调和Adem关系诱导的系统线性依赖严格细化；为从下方界定矩阵秩（即余击中上界），提取显式独立列族：单元素列产生置换子式，无环枢轴系统在所有行序中优化三角子式，$q$支撑列通过超图关联正式定义。关键是，我们识别出一个同余族，其在$\boldsymbol{\textit{F}}_2$上精确分解为单纯形边界矩阵，得到精确的闭式秩公式。所得双侧界对所有$k \boldsymbol{\textit{≥}} 1$和$d \boldsymbol{\textit{≥}} 0$均普遍可计算，这些结果确立了纯组合方法解决击中问题的绝对极限，明确区分了通用离散认证与表示论及权滤层提供的特定次数解。
英文摘要
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.
Comments19 pages. This revision addresses minor errors in the original manuscript and improves the main results to obtain tighter inequalities. The manuscript's title has been modified to reflect these updates. We welcome constructive comments and feedback
