The “hit” problem of five variables in the generic degree and its application

Abstract

Let Ps:=F2[x1,x2,…,xs] be the graded polynomial algebra over the prime field of two elements, F2, in s variables x1,x2,…,xs, each of degree 1. We are interested in the Peterson “hit” problem of finding a minimal set of generators for Ps as a graded left module over the mod-2 Steenrod algebra, A. For s⩾5, it is still open.In this paper, we study the hit problem of five variables in a generic degree. By using this result, we survey Singer's conjecture [26] for the fifth algebraic transfer in the respective degrees. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer.