The negative answer to Kameko's conjecture on the hit problem

Abstract

The goal of this paper is to give a negative answer to Kameko's conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra P k , considered as a module over the Steenrod algebra A , is dominated by an explicit quantity depending on the number of the polynomial algebra's variables k. The conjecture was shown by Kameko himself for k ⩽ 3 in his PhD thesis in the Johns Hopkins University in 1990, and recently proved by us and Kameko for k = 4 . However, we claim that it turns out to be wrong for any k > 4 . In order to deny Kameko's conjecture we study a minimal set of generators for A -module P k in some so-call generic degrees. What we mean by generic degrees is a bit different from that of other authors in the fields such as Crabb–Hubbuck, Nam, Repka–Selick, Wood …We prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables, k, increases. As an immediate consequence of this inductive formula, we recognize that Kameko's conjecture is no longer true for any k > 4 .