The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

Abstract

Let Pk be the graded polynomial algebra F2[x1,x2,…,xk] with the degree of each generator xi being 1, where F2 denote the prime field with two elements.The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra A. Equivalently, we want to find a vector space basis for F2⊗APk in each degree.In this paper, we study a generating set for the kernel of Kameko's squaring operation Sq˜⁎0:F2⊗APk⟶F2⊗APk in a so-called generic degree. By using this result, we explicitly compute the hit problem for k=5 in the respective generic degree.