On $n$-Dimensional Sequences. I

TL;DR

This work extends linear recurring sequence theory to $n$-dimensional sequences over a commutative ring by adopting right-shifting in the Laurent-series ambient space and studying the generating function $\Gamma(s)$ alongside the annihilator $\mathrm{Ann}(s)$. It introduces a border partition and a central border polynomial $\beta_0(f,s)$ to decompose $f\cdot\Gamma(s)$ into a sum of border components, with $\beta_0(f,s)$ capturing the part in $P_n$ and the rest forming border Laurent series. For eventually rectilinear (EVR) sequences, the paper derives ideal-quotient descriptions of $\mathrm{Ann}(s)$, reduces computations to elmination ideals in each variable, and provides algorithms to compute monic generators and bases, particularly over potential (factorial) domains. The findings unify $1$-D and multi-dimensional cases, connect to existing decoding and algebraic coding theory, and yield practical methods for constructing bases of annihilators and related objects in multidimensional settings. These results have applications to multidimensional cyclic codes and related decoding problems, and they offer division-free realizations and complexity bounds in fields.

Abstract

Let $R$ be a commutative ring and let $n \geq 1.$ We study $Γ(s)$, the generating function and Ann$(s)$, the ideal of characteristic polynomials of $s$, an $n$--dimensional sequence over $R$. We express $f(X_1,\ldots,X_n) \cdot Γ(s)(X_1^{-1},\ldots ,X_n^{-1})$ as a partitioned sum. That is, we give (i) a $2^n$--fold ``border'' partition (ii) an explicit expression for the product as a $2^n$--fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is $β_0(f,s)$, the ``border polynomial'' of $f$ and $s$, which is divisible by $X_1\cdots X_n$. We say that $s$ is {\em eventually rectilinear} if the elimination ideals Ann$(s)\cap R[X_i]$ contain an $f_i(X_i)$ for $1 \leq i \leq n$. In this case, we show that $\mbox{Ann}(s)$ is the ideal quotient $(\sum_{i=1}^n(f_i)\ :\ β_0(f,s)/(X_1\cdots X_n)).$ When $R$ and $R[[X_1,X_2, \ldots ,X_n]]$ are factorial domains (e.g. $R$ a principal ideal domain or ${\Bbb F}[X_1,\ldots,X_n]$), we compute {\em the monic generator} $γ_i$ of $\mbox{Ann}(s) \cap R[X_i]$ from known $f_i \in \mbox{Ann}(s) \cap R[X_i]$ or from a finite number of $1$--dimensional linear recurring sequences over $R$. Over a field ${\Bbb F}$ this gives an $O(\prod_{i=1}^n δγ_i^3)$ algorithm to compute an ${\Bbb F}$--basis for $\mbox{Ann}(s)$.