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

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.