The Unique Distribution of Zeros in Compressing Sequences Derived from Primitive Sequences over Z/(pe)

Let $\pmb{Z}/(\pmb{p}^{e})$ be the integer residue ring with odd prime $\pmb{p}$ and integer $\pmb{e}\geq 3$. Any sequence $\underline{\pmb{a}}$ over $\pmb{Z}/(\pmb{p}^{e})$ has a unique p-adic expansion $\underline{\pmb{a}}=\underline{\pmb{a}}_{0}+\underline{\pmb{a}}_{1}\cdot p+\cdots+\underline{\pmb{a}}_{e-1}\cdot p^{e-1}$, where $\underline{\pmb{a}}_{i}$ can be regarded as a sequence over $\pmb{Z}/(\pmb{p})$ for $0\leq i\leq e-1$. Let $\pmb{f}(\pmb{x})$ be a strongly primitive polynomial over $\pmb{Z}/(\pmb{p}^{e})$ and let $\underline{\pmb{a}}, \underline{\pmb{b}}$ be two primitive sequences generated by $\pmb{f}(\pmb{x})$ over $\pmb{Z}/(\pmb{p}^{e})$. Assume $\varphi(\pmb{x}_{0}, \cdots, \pmb{x}_{e-1})=\pmb{x}_{e-1}+ \boldsymbol{\eta}(\pmb{x}_{0}, \cdots, \pmb{x}_{e-2})$, where the degree of $\pmb{x}_{e-2}$ in $\boldsymbol{\eta}(\pmb{x}_{0}, \cdots, \pmb{x}_{e-2})$ is less than $\pmb{p}-1$. It is shown that if $\boldsymbol{\varphi}(\pmb{a}_{0}(t), \cdots, \pmb{a}_{e-1}(\pmb{t}))=0$ if and only if $\boldsymbol{\varphi}(\pmb{b}_{0}(\pmb{t}), \cdots, \pmb{b}_{e-1}(\pmb{t}))=0$ for all nonnegative integer $\pmb{t}$ with $\boldsymbol{\alpha}(\pmb{t})\neq 0$, where $\underline{\boldsymbol{\alpha}}$, is an m-sequence determined by $\pmb{f}(\pmb{x})$ and $\underline{\pmb{a}}_{0}$, then $\underline{\pmb{a}}=\underline{\pmb{b}}$. In particular, when $\boldsymbol{\eta}(\pmb{x}_{0} \cdots, \pmb{x}_{e-2})=0$, it is just the former result on the unique distribution of zeros in the highest level sequences.