Two-Sided Dimension Bounds for the Peterson Hit Problem via Projections and Matrix Minors

Abstract

Let ${\mathcal{P}}_k={\mathbb{F}}_2[x_1,\dots ,x_k]$ 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 $k\ge 1$ 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.

arXiv comment19 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