On n-Dimensional Sequences. I

Let R be a commutative ring and let n ≥ 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(X₁,…,X_n) · Γ(s)(X^-1₁,…,X^-1_n) as a partitioned sum. That is, we give (i) a 2ⁿ-fold "border" partition (ii) an explicit expression for the product as a 2ⁿ-fold sum; the support of each summand is contained in precisely one member of the partition. A key summand is βo(f, s), the "border polynomial" of f and s, which is divisible by X₁ … X_n.We say that s is eventually rectilinear if the elimination ideals Ann(s)∩RX_i] contain an f_i (X_i) for 1 ≤ i ≤ n. In this case, we show that Ann(s) is the ideal quotient ($\text{∑}{}^{\mathrm{n}}{}_{\mathrm{i}}\text{=1(fi) : βo(f, s)/(X}{}_1\text{… X}{}_{\mathrm{n}}\text{)}$).When R and RX₁, X₂ ,…, X_n]] are factorial domains (e.g. R a principal ideal domain or F X₁,…, X_n]), we compute the monic generator γ_i of Ann(s) ∩ RX_i] from known f_i ? Ann(s) ∩ RX_i] or from a finite number of 1-dimensional linear recurring sequences over R. Over a field F this gives an O($\text{∏}{}_{\mathrm{n}}{}^{\mathrm{i}}\text{=1 δγ}{}^3{}_{\mathrm{i}}$) algorithm to compute an F-basis for Ann(s).