Counting polynomials with distinct zeros in Galois ring GR(p2,r)

Abstract

Let $R=GR(p^2,r)$ be a Galois ring of characteristic $p^2$ with cardinality $p^{2r}$, where p is a prime, and ξ is a root of a basic irreducible polynomial of degree r in $Z_{p^2}\left[x\right]$. Let k be a positive integer and m be a integer such that $k+m⩾0$. Fix a polynomial $u\left(x\right)\in R\left[x\right]$ of degree $k+m$. For a subset D of R, let $N_D\left(u\right(x),s)=♯\left\{g\right(x)\in R[x\left]\right|\mathrm{deg}\left(g\right)⩽k-1,u\left(x\right)+g\left(x\right)\text{has exactly s distinct roots in D}\}$. In this paper, we obtain formulae for $N_D\left(u\right(x),s)$ when $D=〈\xi 〉\bigcup \left\{0\right\}$ and $m⩽1$ and give some identities for $N_D\left(u\right(x),s)$ by using the generalization of the Macilliams identity.