Hit polynomials and excess in the \(\bmod p\) Steenrod algebra (Q2747924)

Let \(p\) be an odd prime, \(A\) be a mod.\(p\) Steenrod algebra and \(P[k,i]= P(p^{k-1}i) P(p^{k-2}i) \dots P(i)\), \(k\geq 1\), \(\ell\geq 0\) \((P(i)\) denotes the Steenrod reduced power of degree \(2i(p-1))\). The aim of this paper is to compute the excess of the operation \(\chi(P [k,i])\) where \(\chi: A\to A\) is the canonical anti-automorphism of \(A\). As a consequence of this computation, the author gives sufficient conditions for an element in the polynomial algebra \(P_n=\mathbb{F}_p [v_1,\dots, v_n]\cong H^*((\mathbb{C} p^\infty)^n; \mathbb{F}_p)\) to be in the image of the natural action of \(A\) on \(P_n\). (For the mod.2 analogue of this paper see [\textit{J. H. Silverman}, Math. Proc. Camb. Philos. Soc. 123, No. 3, 531-547 (1998; Zbl 0936.55010)]).