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Dang Võ Phúc

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Preprint Jul 2026

The rank-five Peterson hit problem, the fifth Singer transfer, and a geometric generator in unoriented cobordism

The Peterson hit problem seeks a minimal set of generators for the polynomial algebra $P_s = \mathbb{F}_2[x_1,\dots,x_s]$ as an unstable module over the mod-2 Steenrod algebra $\mathcal{A}$. For rank five, general admissible bases fail, and the interplay between Kameko periodicity and modular invariants becomes computationally complex. In this paper, we study the rank-five cohit module in the generic family $N_d = 27\cdot 2^d - 5$. Exact sparse elimination in degree $49$ processes $292825$ monomials, yielding a hit rank of $289969$ and a cohit dimension of $2856$. We determine the exact weight summands and prove that the weight-$(3,3,2,2,1)$ summand is exactly the kernel of Kameko's operation, with dimension $1891$. These exact values systematically correct the corresponding rank-five kernel and dimension assertions in Nguyen Khac Tin's previous paper. An exact invariant calculation shows that the general linear group invariants in degree $49$ form a one-dimensional space generated by a $283$-term polynomial, and we prove that the fifth Singer cohomological transfer is an isomorphism in this family. Geometrically, the Hilbert-Poincare series of the unoriented cobordism ring gives the dimension of the degree-$49$ cobordism group as $5692$. We prove that the Milnor hypersurface $H_{2,48} \subset \mathbb{R}P^2 \times \mathbb{R}P^{48}$ represents the unique nonzero indecomposable class by computing a tangential Stiefel-Whitney number, providing an explicit geometric generator. However, the evident map from $H_{2,48}$ to the classifying space $B(\mathbb{Z}/2)^5$ sends its fundamental class to a homology class with nonzero $Sq^2_*$. Consequently, this geometric generator cannot be identified with the functional dual of the algebraic invariant, establishing a precise boundary between the Steenrod-theoretic invariant line and the geometric cobordism generator.

Dang Võ Phúc · 0 citations

Preprint Jul 2026

Combinatorial Bounds on the Peterson Hit Problem via Certified Matrix Minors

The Peterson hit problem seeks a minimal set of generators for the polynomial algebra $\mathcal P_k=\mathbb F_2[x_1,\ldots,x_k]$ as a module over the mod--2 Steenrod algebra. While completely resolved for $k \leq 4$, the unrestricted problem remains widely open for $k \geq 5$, where the combinatorial explosion of basis elements renders exact algorithmic computation intractable. To bypass full Gaussian elimination, we model the degree--$d$ hit space via a sparse matrix driven by the Cartan formula and Lucas's theorem, shifting the focus to the construction of certified matrix minors. We first prove that strict spike monomials exactly characterize the zero rows, establishing a hard structural limit on coordinate-level annihilators. To bound the matrix rank from above (cohit lower bound), we derive exact zero-column formulae, which are strictly refined by the exact homology of the $\operatorname{Sq}^1$-layer and systematic linear dependencies induced by Adem relations. To bound the rank from below (cohit upper bound), we extract explicit independent column families: singleton columns yield permutation minors, acyclic pivot systems optimize triangular minors across all row orders, and $q$-support columns are formalized through hypergraph incidence. Crucially, we identify a congruence family that decomposes precisely into simplicial boundary matrices over $\mathbb F_2$, yielding a sharp closed-form rank formula. The resulting two-sided bounds are universally computable for every $k \geq 1$ and $d \geq 0$. Significantly, these results establish the absolute limits of purely combinatorial approaches to the hit problem, cleanly separating universal discrete certificates from the degree-specific resolutions provided by representation theory and weight filtrations.

Dang Võ Phúc · 0 citations