Geodesic
Polynomial transformations of linear recurrent sequences over finite commutative rings
Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 3-36

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Let $u$ be a linear recurring sequence (LRS) over a finite commutative local ring $R$ with identity, and let $\Phi(x)\in R[x]$. We find a characteristic polynomial $H(x)$ and prove an upper estimate for the rank (linear complexity) over $R$ of the sequence $v=\Phi(u)$. If $\bar u$ is an $m$-sequence over the residue field $\bar R=R/J(R)=GF(q)$ of the ring $R$ and $\deg\Phi(x)\le q-1$, then this estimate is attained and $H(x)$ is a minimal polynomial of $v$. Analogous results are obtained for the sequence $v=\Phi(u_1, \ldots, u_K)$ which is a polynomial transform of $K$ linear recurrences $u_1, \ldots, u_K$ over $R$. 
@article{DM_2000_12_3_a0,
     author = {V. L. Kurakin},
     title = {Polynomial transformations of linear recurrent sequences over finite commutative rings},
     journal = {Diskretnaya Matematika},
     pages = {3--36},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2000_12_3_a0/}
}
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V. L. Kurakin. Polynomial transformations of linear recurrent sequences over finite commutative rings. Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 3-36. http://geodesic.mathdoc.fr/item/DM_2000_12_3_a0/