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

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 $h\ge 6$, 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.

104 pages. Update the appendix with the complete admissible monomial basis of degree 26 in $P_6$ and some ${\Sigma }_6$-invariants of $Q{P}_{26}^{\otimes 6}$ associated with certain weight vectors