Decomposition of tableaus annihilated by zero-dimensional ideals

A (1-dimensional) linear recursive sequence s over a field K decomposes in the following canonical way. Let I K[x] be the annihilator ideal of s. Since K[x] is a principal ideal domain, I=fK[x] for some polynomial f, which can be factored as f1 fr where the fi are coprime. Thus s can be uniquely written as a sum of sequences si having annihilator ideals Ii=fiK[x]. Furthermore, each fi is a power of an irreducible polynomial fi=(gi)ei. Each sequence si can be uniquely written as a ei-fold sum of pointwise products of a “binomial” sequence with a sequence annihilated by gi. Finally, a sequence annihilated by an irreducible polynomial gi is given by a trace formula. See, for instance, [N. Zierler, W.H. Mills, J. Algebra 27 (1973) 147–157]. We show that a completely analogous decomposition (which subsumes the 1-dimensional case) holds for n-dimensional linear recursive sequences, i.e., tableaus annihilated by zero-dimensional ideals of K[x1,…,xn].