# Decomposition of tableaus annihilated by zero-dimensional ideals

## Abstract

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].

## Department(s)

Mathematics

## Document Type

Article

## DOI

https://doi.org/10.1016/s0021-8693(03)00417-4

## Publication Date

2003

## Recommended Citation

Fu, David, Mark Heiligman, and Cameron Wickham. "Decomposition of tableaus annihilated by zero-dimensional ideals." Journal of Algebra 267, no. 2 (2003): 404-420.

## Journal Title

Journal of Algebra