On n-Dimensional Sequences I

Let R be a commutative ring and let n ? 1. We study ?(s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an n-dimensional sequence over R .We express f(X1,?,Xn) · ?(s)(X-11,?,X-1n) 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 so(f, s), the "border polynomial" of f and s, which is divisible by X1 ? Xn.We say that s is eventually rectilinear if the elimination ideals Ann(s)?RXi] contain an fi (Xi) for 1 ? i ? n. In this case, we show that Ann(s) is the ideal quotient (?ni=1(fi) : so(f, s)/(X1 ? Xn )).When R and RX1, X2 ,?, Xn]] are factorial domains (e.g. R a principal ideal domain or FX1,?, Xn]), we compute the monic generator ?i of Ann(s) ? RXi] from known fi ? Ann(s) ? RXi] or from a finite number of 1-dimensional linear recurring sequences over R. Over a field F this gives an O(?ni=1 ??3i) algorithm to compute an F-basis for Ann(s).