Abstract. We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k := F 2 [x 1 , x 2 , . . . , x k ] as a module over the mod-2 Steenrod algebra, A. In this paper, we study a minimal set of generators for A-module P k in some so-called generic degrees and apply these results to explicitly determine the hit problem for k = 4. Dedicated to Prof. N. H. V. Hưng on the occasion of his sixtieth birthday Introduction and statement of resultsLet V k be an elementary abelian 2-group of rank k. Denote by BV k the classifying space of V k . It may be thought of as the product of k copies of the real projective space RP ∞ . ThenHere the cohomology is taken with coefficients in the prime field F 2 of two elements.Being the cohomology of a space, P k is a module over the mod-2 Steenrod algebra A. The action of A on P k is explicitly given by the formulaand subject to the Cartan formulafor f, g ∈ P k (see Steenrod and Epstein [30]). A polynomial f in P k is called hit if it can be written as a finite sum f = i>0 Sq i (f i ) for some polynomials f i . That means f belongs to A + P k , where A + denotes the augmentation ideal in A. We are interested in the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k as a module over the Steenrod algebra. In other words, we want to find a basis of the F 2 -vector spaceThe hit problem was first studied by Peterson [22,23]

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