On the dimensions of the graded space 𝔽2 ⊗𝒜𝔽2[x1,x2,…,xs] at degrees s + 5 and its relation to algebraic transfers

For the purposes of our analysis, we shall work with the Steenrod algebra [Formula: see text] over the field of two elements, [Formula: see text] Our paper focuses on expanding the outcomes from our preceding study, which serves as Part I. Precisely, we will initially compute the dimension of the “cohit” space [Formula: see text], where [Formula: see text] is defined as the graded polynomial ring [Formula: see text], for the specific case of [Formula: see text]. This computation will be carried out explicitly for the generic degree [Formula: see text], where [Formula: see text] is a positive integer. We have developed a novel algorithm implemented in SageMath to compute the case where [Formula: see text] We subsequently study the dimension of [Formula: see text] in degrees [Formula: see text] for [Formula: see text] One of our findings is the correction of some erroneous results in the previous work by Moetele and Mothebe. We, additionally, give an explicit formula for the dimension of [Formula: see text] in degree fourteen for all [Formula: see text] and in degree fifteen for all [Formula: see text] We also describe a computational method implemented in SageMath to explicitly account for the number of spikes of degree [Formula: see text] in the [Formula: see text]-module [Formula: see text], applicable for all positive integers [Formula: see text] In application, we examine W. Singer’s conjecture on the algebraic transfer of rank [Formula: see text] in degrees [Formula: see text] for all [Formula: see text] and of ranks [Formula: see text] in internal degrees [Formula: see text] ranging from 13 to 15. Considering higher homological degrees, we analyze the behavior of the algebraic transfer of rank 7 in the generic degrees [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] We then demonstrate that the (in)decomposable elements [Formula: see text] [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are not in the image of the Singer transfer.