On the Action of Steenrod Operations on Polynomial Algebras

Authors: İ. KARACA

Abstract: Let \( \bba \) be the mod-\( p \) Steenrod Algebra. Let \( p \) be an odd prime number and \( Z_{p} = Z/pZ \). Let \( P_{s} = Z_{p} [x_{1},x_{2},\ldots,x_{s}]. \) A polynomial \( N \in P_{s} \) is said to be hit if it is in the image of the action \( A \otimes P_{s} \ra P_{s}. \) In [10] for \( p=2, \) Wood showed that if \( \a(d+s) > s \) then every polynomial of degree \( d \) in \( P_{s} \) is hit where \( \a(d+s) \) denotes the number of ones in the binary expansion of \( d+s \). Latter in [6] Monks extended a result of Wood to determine a new family of hit polynomials in \( P_{s}. \) In this paper we are interested in determining the image of the action \( A\otimes P_{s} \ra P_{s} \). So our results which determine a new family of hit polynomials in \( P_{s} \) for odd prime numbers generalize cononical antiautaomorphism of formulas of Davis [2], Gallant [3] and Monks [6].

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