Decomposition of tableaus annihilated by zero-dimensional ideals

✍ Scribed by David Fu; Mark Heiligman; Cameron Wickham

• Book ID: 104140846
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 161 KB
• Volume: 267
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00417-4

✦ Synopsis

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 = f K[x] for some polynomial f , which can be factored as f 1 • • • f r where the f i are coprime. Thus s can be uniquely written as a sum of sequences s i having annihilator ideals

Furthermore, each f i is a power of an irreducible polynomial f i = (g i ) e i . Each sequence s i can be uniquely written as a e i -fold sum of pointwise products of a "binomial" sequence with a sequence annihilated by g i . Finally, a sequence annihilated by an irreducible polynomial g i 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[x 1 , . . . , x n ].