INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/(p~e)

摘要

Let Z/(p~e) be the integer residue ring modulo p~e with p an odd prime and integerrne > 3. For a sequence a over Z/(p~e), there is a unique p-adic decomposition a = a_0 + a_1·p + … +rna_(e-1)·p~(e-1), where each a_i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(p~e) and G'(f(x),p~e) the set of all primitive sequences generated by f(x) over Z/(p~e). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(l + deg(μ(x)), p- 1) = 1, setrnφ_(e-1)(x_0, x_1 ,…, x_(e-1) = x_(e-1)·[μ(x_(e-2) + η_(e-3)(x_0, x_1, …, x_(e-3))] + η_(e-2)(x_0, x_1, …, x_(e-2)), rnwhich is a function of e variables over Z/(p). Then the compressing maprnφ_(e-1) : G'(f(x), p~e)→(Z/(p))~∞, a→φ_(e-1)(a_0, a_1, …, a_(e-1))rnis injective. That is, for a,b ∈ G'(f(x),p~e), a = b if and only if φ_(e-1)(a_0, a_1, …, a_(e-1)) = φ_(e-1)(b_0, b_1,…, b_(e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions φ_(e-1) and ψ_(e-1) over Z/(p) are both of the above form and satisfyrnfor a,b ∈ G'(f(x),p~e), the relations between a and b, φ_(e-1) and φ_(e-1) are discussed.