On the hit problem for the polynomial algebra and the algebraic transfer

TL;DR

This paper advances understanding of Singer's conjecture by analyzing the cohit module for specific degrees and values of h, providing new dimensional results and identifying elements outside the algebraic transfer's image, verified computationally.

Contribution

It extends previous work on the hit problem, establishing new dimensional results for the cohit module and demonstrating non-membership of certain elements in the algebraic transfer's image.

Findings

Dimension of cohit module matches the order of a specific factor group for h≥6.

Certain non-zero elements are proven not to be in the image of the algebraic transfer.

Results verified using the OSCAR computer algebra system.

Abstract

This paper investigates Singer's conjecture by examining the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ for specific degrees and values of $h$. Utilizing hit problem techniques, we extend previous work by Mothebe et al. and establish key dimensional results. Notably, for $h\geq 6$, we prove that the cohit module's dimension in certain degrees matches the order of a specific factor group. Our contributions include demonstrating that certain non-zero elements do not belong to the image of the Singer algebraic transfer. All results were verified using the OSCAR computer algebra system.