Further results on the compression maps of primitive sequences over Z=(pe)
Chao LI, Longjiang Qu, Hai Jun Xiong · Scientia Sinica Mathematica · 2014
Let Z/(pe) be the integer residue ring with p prime and e ≥ 2. Let f(x) be a primitive polynomial over Z/(pe) with degree n and G(f(x), pe) the set of all primitive sequences over Z/(pe) generated by f(x). For any sequence a∈G(f(x), pe), it has a unique p-adic expansion a = a0 + a1p +…+ ae-1pe-1. Let ø(x0, x1,…, xe-1) = g(xe-1)+μ(x0, x1,…, xe-2) be a function from Fpe to Fp. Then ø can induce a compression mapping from G(f(x), pe) to Fp∞. In recent years, Zhu, Tian and Qi have proved that the compression mapping is injective under the condition that p is an odd prime and f(x) is a strongly primitive polynomial. In this paper, we improve their results. More exactly, we prove that the compression mapping is also injective only under the condition that p is an odd prime and f(x) is a primitive polynomial. Of course we should ask that deg(g(x)) is an odd number or g(x) = xk +Σi=0k-2 cixi.