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