Dimension Polynomials in Difference and Difference-Differential Algebra

Abstract

Let R be a difference ring with a basic set σ = {α _l,...,α _n } and let T = T _σ be a free commutative semigroup generated by the elements α _l,...,α _n . As in Section 3.3, by the order of an element $\tau = \alpha _1^{{k_1}}...\alpha _n^{{k_n}} \in T\left( {{k_1},...,{k_n} \in } \right)$ we shall mean the number ord $\tau = \Sigma _{i = 1}^n{k_1}$ and set T _r = {τ ∈ T | ord τ = r}, T(r) = {τ ∈ T | ord τ ≤ r} for any r ∈ ⑅. Furthermore, let U be a ring of difference (σ-) operators over the ring R. As in Chapter 3, if $f = \sum\nolimits_{\tau \in T} {{a_\tau }\tau } \in U$ (a _τ τ ∈ R for any τ ∈ T and a _τ = 0 for almost all τ ∈ T), then the number ord f = max{ord τ | a _τ ≠ 0} will be called the order of the element f.