Year: 2026
Type: preprint
Abstract: Let $\mathcal P_k=\mathbb F_2[x_1,\ldots,x_k]$, with $\deg x_i=1$, be the polynomial algebra equipped with the standard unstable action of the mod-$2$ Steenrod algebra $\mathcal A$. The Peterson hit problem asks for a minimal homogeneous set of $\mathcal A$-generators for $\mathcal P_k$, or equivalently for a homogeneous basis of the cohit space $Q\mathcal P_k=\mathcal P_k/\mathcal A^{+}\mathcal P_k = \mathbb F_2\otimes_{\mathcal A}P_k$, where $\mathcal A^{+}$ denotes the augmentation ideal of $\mathcal A$. The dimension of $(Q\mathcal P_k)_d$ therefore gives the number of generators required in positive degree $d$. Although solved in every degree $d$ for $k\leq 4,$ the problem remains open in general. As the number of variables $k$ and degrees $d$ increase, exact calculations involve increasingly large systems of Steenrod relations. Motivated by this difficulty, we establish explicit upper and lower bounds for $\dim_{\mathbb F_2}(Q\mathcal P_k)_d$ for arbitrary positive integers $k$ and $d$. Our method combines binary combinatorics and linear algebra with graph-theoretic and simplicial structures in the matrix of the generating Steenrod squares. We characterize its zero rows, count its zero columns, and refine rank estimates using evaluated Adem relations. Triangular minors are described by uniquely restricted matchings, while selected column families form augmented simplicial boundary matrices. An explicit projection modulo $\operatorname{im}\operatorname{Sq}^1$ separates the known first-square rank from the rank of a smaller matrix. We classify the zero rows of this projected matrix and derive a closed formula for their number. Pullbacks of the corresponding coordinate functionals form a linearly independent family in the classical dual-spike module, containing all coordinate spikes and, in some degrees, additional non-coordinate functionals. Selected minors yield upper cohit bounds, whereas these functionals, evaluated Adem relations, and spike-insertion constructions yield lower bounds. The resulting inequalities do not require a complete cohit basis or the full hit rank.
Source: arXiv (Cornell University)
Authors: Phuc, Dang Vo, Danchev, Peter V.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213500657