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A note on the hit problem for the polynomial algebra in the case of odd primes and its application

Dang Vo Phuc

TL;DR

This work advances the odd-prime hit problem for $P_h=\mathbb F_p[t_1,...,t_h]$ by developing a $p$-adic digit framework, height-tail analysis, and Cartan–lex ordering to dissect $\mathscr A_p$-hits and invariants. It proves that for rank $h=3$ and generic degrees $n^{(t)}$ ($t=1,...,4$), the third Singer transfer $Tr_3^{\mathscr A_p}(\mathbb F_p)$ is an isomorphism, using a reduction to torus invariants on associated graded pieces and a one-dimensional invariant line in the top exterior slice. The paper combines rigorous algebraic results with computational verification (OSCAR and SageMath) to obtain explicit invariant bases and dimension counts, and to illustrate the graded vs full Cartan form differences. These results deepen understanding of Ext groups via the algebraic transfer and provide concrete tools for analyzing $GL(3,\mathbb F_p)$-invariants in the hit problem for odd primes.

Abstract

Let $P_h = \mathbb{F}_p[t_1,\dots,t_h]$ be the polynomial algebra over $\mathbb{F}_p$ ($p$ prime). We consider the hit problem: finding a minimal generating set for $P_h$ as a module over the mod $p$ Steenrod algebra $\mathscr{A}_p$, or equivalently, determining a basis for $\mathbb{F}_p \otimes_{\mathscr{A}_p} P_h$. This problem is related to the $\mathscr{A}_p$-module structure of $H^*(V; \mathbb{F}_p) \cong Λ(V^\sharp) \otimes P_h$, where $V$ is an elementary abelian $p$-group of rank $h$. Information about the hit problem aids in studying the Singer algebraic transfer $Tr_h^{\mathscr{A}_p}$, a homomorphism from $GL(h, \mathbb{F}_p)$-coinvariants related to $H^*(V; \mathbb{F}_p)$ to ${\rm Ext}_{\mathscr{A}_p}^{h,h+*}(\mathbb{F}_p, \mathbb{F}_p)$, which helps analyze Ext groups. This work studies $\mathscr{A}_p$-generators for $P_h$ when $p$ is an odd prime. As an application, we investigate the third algebraic transfer ($h=3$) in certain generic degrees. Our main result shows that this transfer is an isomorphism in these degrees.

A note on the hit problem for the polynomial algebra in the case of odd primes and its application

TL;DR

This work advances the odd-prime hit problem for by developing a -adic digit framework, height-tail analysis, and Cartan–lex ordering to dissect -hits and invariants. It proves that for rank and generic degrees (), the third Singer transfer is an isomorphism, using a reduction to torus invariants on associated graded pieces and a one-dimensional invariant line in the top exterior slice. The paper combines rigorous algebraic results with computational verification (OSCAR and SageMath) to obtain explicit invariant bases and dimension counts, and to illustrate the graded vs full Cartan form differences. These results deepen understanding of Ext groups via the algebraic transfer and provide concrete tools for analyzing -invariants in the hit problem for odd primes.

Abstract

Let be the polynomial algebra over ( prime). We consider the hit problem: finding a minimal generating set for as a module over the mod Steenrod algebra , or equivalently, determining a basis for . This problem is related to the -module structure of , where is an elementary abelian -group of rank . Information about the hit problem aids in studying the Singer algebraic transfer , a homomorphism from -coinvariants related to to , which helps analyze Ext groups. This work studies -generators for when is an odd prime. As an application, we investigate the third algebraic transfer () in certain generic degrees. Our main result shows that this transfer is an isomorphism in these degrees.
Paper Structure (11 sections, 17 theorems, 67 equations, 2 tables)

This paper contains 11 sections, 17 theorems, 67 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main theorems and the proofs
  3. $p$--adic digits and distributions.
  4. Height tails.
  5. Cartan--lex Monomial Order.
  6. Appendix
  7. An algorithm in the OSCAR computer algebra system
  8. Some illustrative examples
  9. Additional examples
  10. SageMath Implementation and Verification
  11. Analysis of Output.

Key Result

Theorem 2.1

Let us consider odd primes $p.$ Then, in the $\mathscr A_p$-module $P_1 = \mathbb F_p[t],$ the monomial $t^{d}$ is not in the image of $\mathscr A_p$ if and only if $d$ is of the form $(i+1)p^{k} - 1,$ for some $i,\, k$ satisfying $1\leq i\leq p-1$ and $k\geq 0.$ Consequently, $(\mathbb F_p\otimes_{

Theorems & Definitions (37)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2: cf. Crossley
  • Remark 2.3
  • Theorem 2.4: Theorem 2.5 of the original paper Phuc_original
  • proof
  • Theorem 2.5: Theorem 2.6 of the original paper Phuc_original
  • Theorem 2.6: Theorem 2.7 of the original paper Phuc_original
  • ...and 27 more