On a Construction for the Generators of the Polynomial Algebra as a Module Over the Steenrod Algebra

The hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ and its application

“…In the present study, we investigate the hit problem of five variables in degree n of the form (1) for r = 5, m = 42 and s an arbitrary non-negative integer (i.e., n = 5(2 s − 1) + 42.2 s ). The result confirms Sum's conjecture [15] for the relation between the minimal sets of A 2 -generators of the algebras P t−1 and P t in the case t = 5 and the above generic degree. An efficient approach for solving the hit problem of five variables in this case has been given.…”

“…In this article, we explicitly solve the hit problem of five variables in the "generic" degree n = 5(2 s − 1) + 42.2 s for every non-negative integer s. The result confirms Sum's conjecture [15] for the relation between the minimal sets of A 2 -generators of the algebras P t−1 and P t in the case t = 5 and degree n above. An efficient approach for surveying the hit problem of five variables has been presented.…”

“…A central problem of homotopy theory is to determine a minimal set of generators for the Z 2 -graded vector space {(Z 2 ⊗ A2 P t ) n } n≥0 . It is called the "hit" problem for Steenrod algebra and has been completely solved for t ≤ 4.In this article, we explicitly solve the hit problem of five variables in the "generic" degree n = 5(2 s − 1) + 42.2 s for every non-negative integer s. The result confirms Sum's conjecture [15] for the relation between the minimal sets of A 2 -generators of the algebras P t−1 and P t in the case t = 5 and degree n above. An efficient approach for surveying the hit problem of five variables has been presented.…”

“…It has been surveyed by many authors. (See Kameko [2], Mothebe-Uys [4], the present author [8,9,10,11], Singer [13], Sum [14,15], Wood [17], and others. )…”

“…The structure of QP t was explicitly described by Peterson [5] for t = 1, 2, by Kameko [2] for t = 3, and by Sum [14] for t = 4. When t = 5 and in some generic degrees of the form (1), it has been surveyed by us [6,7,8,9,10,11,15], and Tín [16]. The problem is not yet determined in general, even with the help of the computer.…”